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THE  PUBLIC   SCHOOL 
MENTAL    AEITHMETIC 


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THE  PUBLIC  SCHOOL 

MENTAL    ARITHMETIC 

BASED  ON 
McLELLAN  AND  DEWEY'S  '*  PSYCHOLOGY  OF  NUMBER  " 

BY 

J.  A.  McLELLA]^,  A.M.,  LL.D. 

PRESIDENT  ONTARIO   NORMAL  COLLEGE 

AUTHOR  (with  DR.   DEWEY)   OF    "THE  PSYCHOLOGY  OF  NUMBER,"    "APPLIED 

PSYCHOLOGY,"    "ELEMENTS  OF  ALGEBRA,"   ETC. 

AND 

A.  F.  AMES,  A.B. 

HONOR  GRADUATE   IN  MATHEMATICS  ;  FORMERLY  MATHEMATICAL 

MASTER,    ST.   THOMAS   COLLEGIATE   INSTITUTE 

SUPERINTENDENT   OF   SCHOOLS,    RIVERSIDE,    ILL. 


THE   MACMILLAN   COMPANY 

LONDON:  MACMILLAN  &  CO.,  Ltd. 

1899 

All  rights  reserved 


Copyright,  1899, 
By  the  MACMILLAN  COMPANY. 


Set  up  and  electrotyped  January,  1899.      Reprinted  August, 


Nortoooft  ^ress 

J.  S.  Cushing  &  Co.  —  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


^c^ 


PREFACE 

It  has  recently  been  stated  by  a  well-known  col- 
lege professor  that  "boys  enter  college  or  training 
school  at  eighteen,  after  having  spent  from  one  sixth 
to  one  fourth  of  their  entire  school  life  in  studying 
mathematics.  Yet  they  know  very  little  mathe- 
matics. In  their  examination  the  asking  of  even 
three  questions  shows  that  they  haven't  the  dimmest 
idea  of  what  it  is  all  about."  This  statement,  if 
true,  does  not  prove  —  as  the  professor  seems  to 
think  it  does  —  the  justice  of  the  Hamiltonian  on- 
slaught on  mathematical  study.  It  simply  proves 
that  the  prevailing  methods  of  teaching  arithmetic 
are  radically  wrong.  The  serious  defects  in  existing 
methods  are  mainly  due  to  the  fact  that  they  take 
no  account  of  the  real  nature  of  number,  and  of  how 
the  child's  mind  works  in  grasping  the  concepts  of 
number  and  numerical  relations.  In  other  words, 
arithmetic  has  never  been  "psychologized."  If 
there  is  a  science  of  education  and  rational  methods 
founded  upon  it,  there  must  be  a  psychology  of 
arithmetic,  a  psychology  of  language,  etc.  The  one- 
sided theory  that  education  is  concerned  only  with 
fitting  the  child  for  existing  civilization  has  made 


vi  PREFACE 

the  so-called  "practical"  aims  and  methods  domi- 
nant in  school  work.  These  methods  are,  and  ever 
must  be,  essentially  defective,  inasmuch  as  they  are 
founded  on  a  half  truth;  they  take  no  account  of 
the  powers  and  capacities  of  the  individual  who  is 
to  be  made  an  effective  instrument  in  maintaining 
and  perfecting  this  civilization  into  which  he  is 
born.  The  individual,  indeed,  lives  and  moves  and 
has  his  being  in  society,  and  therefore  there  is  a 
social  side  to  education.  Sociology  must  have  some- 
thing to  say  on  the  problem  of  education.  But,  on 
the  other  hand,  society  lives  and  moves  and  has  its 
being  only  through  the  development  of  the  individ- 
ual ;  that  is  to  say,  education  has  its  psychological 
side.  While  the  social  side,  the  idea  of  fitting  the 
individual  to  play  his  part  in  existing  civilization, 
may  give  a  standard  for  the  development  of  the 
powers  and  capacities  of  the  individual,  and  may 
point  out  the  subject-matter  to  be  used  in  this  devel- 
opment, it  does  not  show  how  this  development  of 
powers,  this  adaptation,  is  to  be  secured.  In  other 
words,  sociology  indicates  what  is  to  be  done  with 
the  individual,  but  fails  to  show  how  it  is  to  be 
done.  This  is  a  question  of  psychology.  The 
meaning  of  this  is  that  we  must  have  both  a  psy- 
chology of  the  individual  mind  and  a  psychology  of 
the  subjects  through  which  it  is  to  be  developed. 

On  this  principle  the  "  Public  School  Arithmetic," 
the  "Primary  Public  School  Arithmetic,"  and  the 


PREFACE  yii 

"  Public  School  Mental  Arithmetic  "  have  been  pre- 
pared. It  is  believed  that,  by  direct  teaching  and 
helpful  suggestion,  these  books  will,  in  some  degree, 
contribute  to  the  growth  of  a  rational,  and  therefore 
economical,  method  of  teaching.  At  all  events,  the 
series  differs  from  all  other  text-books  in  being  based 
on  the  Psychology  of  Number. 

This  "  Mental  Arithmetic"  completes  the  series  and 
completes  the  method.  In  the  methods  of  the  schools 
— speaking  generally — the  ignoring  of  mental  arith- 
metic or  the  teaching  of  it  in  a  haphazard,  and  there- 
fore ineffective,  way  contributes  very  largely  to  the 
present  unsatisfactory  results,  viz.  "little  mathemat- 
ics and  less  training  of  logical  faculty."  Mental 
arithmetic,  systematically  taught  from  a  rationally 
prepared  text-book,  is  the  life  and  soul  of  rational 
method.  There  is  constant  adaptation  to  the  nor- 
mal mental  action  of  the  child.  During  the  les- 
son the  teacher  is  in  vital  touch  with  the  child's 
mind;  sees  the  child's  personal  self-activity  in  the 
making  of  images  and  in  controlling  their  move- 
ments. There  is  hence  the  least  possible  waste  for 
both  teacher  and  pupil.  The  teacher  takes  care  of 
the  image,  and  then  the  concept  takes  care  of  itself. 
From  long  and  varied  experience,  both  in  teaching 
the  subject  and  inspecting  the  teaching  of  others,  it 
is  firmly  held  that,  compared  with  "  written  "  arith- 
metic alone,  mental  arithmetic,  if  systematically 
taught,  will  produce  at  least  twice  the  knowledge 


yiii  PREFACE 

and  twice  tlie  power  in  a  given  time.      The  dis- 
tinguishing features  of  the  book  are : 

1.  It  is  not  a  book  of  puzzles  for  ingenious  analy- 
sis or  of  "conundrums"  —  to  use  a  favorite  question- 
begging  epithet  —  for  the  exercise  of  a  nimble  fancy, 
but  a  book  of  ideas  and  principles  for  easy  mastery 
by  rational  method.  The  subject-matter  and  the 
method  are  one. 

2.  Like  the  two  books  for  written  arithmetic,  the 
"  Mental  Arithmetic  "  is  based  on  the  idea  that  num- 
ber is  the  tool  of  measurement,  and  that  measurement 
takes  its  rise  in  human  activity  satisfying  human 
needs.  Therefore  all  the  processes  have  meaning 
for  the  pupil.  They  are  connected  with  his  own 
experience.  The  child,  in  the  first  lesson  and  in  all 
lessons,  is  always  learning  with  what  he  has  learned  ; 
a  primary  idea  unifying  all  and  throwing  light  on 
all.  This  means  that  there  is  continuity.  The  very 
first  question  looks  towards  the  very  last.  Every 
"  new  rule  "  deepens  interest ;  for  it  is  but  common- 
sense  application  of  a  certain  numerical  hahit  to 
slightly  novel  conditions  —  to  the  acquisition  of  a 
new,  but  related,  habit. 

3.  It  keeps  constantly  in  view  the  value  of  the 
image  —  imaging  quantity  and  quantity  relations. 
There  is  constant  appeal  to  the  child's  imagination, 
and  not  merely  to  abstract  reasoning  power. 

4.  The  ihciportant  idea  of  "balance  or  equation" 
is  frequently  stated  —  made  familiar  to  the  pupil  — 


PREFACE  ix 

for  the  insight  which  it  gives  into  the  problem  and 
its  solution.  This  idea  of  balance,  which  results 
from  measurement,  is  present  in  every  problem  in 
arithmetic.  Its  recognition  by  the  pupil  is  essential 
to  the  economical  solution  of  the  problem.  See 
"  Psychology  of  Number,"  p.  41. 

5.  There  is  constant  insistence  on  the  clear  appre- 
hension and  statement  of  the  elements  of  the  ques- 
tion. This  is  fundamental  in  the  intelligent  handling 
of  problems  ;  and  the  teacher  should  frequently  test 
the  pupil  to  see  if,  after  reading  the  problem,  he 
knows  what  is  in  it,  and  to  train  him  to  pay  careful 
attention  to  what  he  reads. 

6.  From  the  gradual  psychological  development 
of  the  subject  the  method  is  given  in  the  presentation 
of  the  matter.  Matter  and  method  and  process  of 
education  are  correlated.  The  teacher  has  not  to 
trouble  himself  with  books  and  articles  on  methods 
and  devices.  P^very  principle  and  process  is  pre- 
sented as  the  natural  movement  of  the  mind  de- 
mands. The  best  methods  are  followed  by  the  best 
results  ;  for  matter,  method,  and  results  are  a  unity. 

7.  Number  concepts  are  of  gradual  growth.  The 
book  is  constructed  to  promote  in  the  best  way  this 
normal  growth.  To  secure  this  with  the  greatest 
certainty  and  economy  all  the  questions  and  prob- 
lems in  the  book  are  original. 

It  is  hoped  that  the  teacher  will  find  the  language 
which  Dr.  Dewey  has  used  in  characterizing  '•'-  The 


X  PREFACE 

Primary  Public  School  Arithmetic "  equally  appli- 
cable to  the  Mental  Arithmetic.  He  says,  "  It  would 
be  difficult,  if  not  impossible,  to  make  a  book  which 
would  start  more  fully  from  what  is  within  the  nat- 
ural range  of  children's  experience  and  capacity. 
The  book  clearly  represents  in  every  lesson  a  care- 
ful study  of  children  as  well  as  of  arithmetic.  In 
addition  to  this  the  development  of  the  subject- 
matter  is  so  carefully  thought  out,  the  transition 
from  one  topic  to  another  so  thoroughly  attended 
to,  that  I  believe  the  book,  in  the  hands  of  a  rea- 
sonably good  teacher,  will  not  only  give  thorough 
knowledge  of  all  the  numerical  combinations  and 
principles  employed  in  this  period  of  school  life,  but, 
what  is  even  more  important,  will  develop  a  num- 
ber sense,  a  capacity  to  understand  and  apply  intel- 
ligently what  is  learned.  I  can  easily  believe  that 
this  book  will  make  a  delight  to  many  children  of 
what  otherwise  would  be  a  bugbear." 


CONTENTS 

PAGE 

Quantity  —  Unit  —  Number  .        .        .        .        .        .  1 

Addition 7 

Subtraction .        .        .18 

Multiplication 26 

Division 39 

Square  Root 49 

Greatest    Common    Measure    and    Least    Common 

Multiple 52 

Fractions 56 

Addition  and  Subtraction  of  Fractions  ...  74 

Multiplication  and  Division  of  Fractions      .        .  81 

Decimals 95 

Compound  Quantities 99 

Percentage  and  its  Applications       .        .        .        .  109 

Miscellaneous  Examples 127 

Multiplication  Table  —  Table  of  Compound  Quan- 
tities    134 

xi 


MENTAL  ARITHMETIC 


oXXa 


QUANTITY  —  UNIT  —  NUMBER 

Lesson  1 

1.  Make  a  mental  picture  of  a  line  12  in.  long 
and  also  of  a  unit  of  measure  3  in.  long.  How  many 
times  will  the  3-in.  unit  measure  the  quantity  12  in.? 

2.  The  number  4  obtained  in  example  1  is  4  what? 
4  units  of  3-in.  in  what  ? 

3.  This  number  4  is  often  called  the  ratio  oi  12 
in.  to  8  in.     What  is  the  ratio  of  12  in.  to  4  in.  ? 

4.  What  part  of  12  in.  is  equal  to  3  in.  ?  ^  is 
often  called  the  ratio  of  3  in.  to  12  in.  What  is  the 
ratio  of  4  in.  to  12  in.  ? 

5.  Find  the  number  of  8-in.  units  of  length  in 
each  of  these  quantities  :  15  in. ;  2  ft. ;  30  in. ;  1  yd. ; 
1  ft.  6  in. ;  2  ft.  3  in.  What  is  the  ratio  of  each  of 
these  quantities  to  the  unit  ?  Of  the  unit  to  each 
quantity  ? 

6.  In  question  5  how  did  you  find  the  number  of 
8-in.  units  in  each  quantity  ? 

B  1 


2  MENTAL   ARITHMETIC 

7.  How  would  you  find  the  number  of  2-in.  units 
in  a  given  quantity  ?  Of  3-in.  units  ?  Of  4-in. 
units  ?     Of  units  of  any  length  ? 

8.  Find  the  number  of  2-in.  units  in  each  of 
these  quantities:  14  in.,  1  ft.  8  in.,  24  in.;  2  ft. 
6  in.;  1yd.  4  in.;  1yd.  1ft. 

9.  In  questions  10-15  name  the  quantity  and 
the  unit  in  each  case.  How  will  you  find  the  num- 
ber of  units  in  each  quantity  ? 

10.  Into  how  many  hair  ribbons,  each  9  in.  long, 
can  you  divide  a  piece  36  in.  ? 

11.  A  can  contains  12  qt.  of  milk.  How  many 
pitchers,  each  containing  3  pt.,  can  be  filled  with  its 
contents  ? 

12.  A  furnace  burns  on  the  average  5  T.  of  coal 
a  month.     In  how  many  months  will  it  burn  40  T.  ? 

13.  How  many  pounds  of  sugar  at  6^  a  lb.  will 
cost  48  ^  ?  How  many  pounds  of  sugar  at  5  ^  a  lb. 
will  cost  I  2  ? 

14.  How  many  barrels  of  flour  at  f  5  a  bbl.  will 
cost  1 45? 

15.  How  many  pounds  of  raisins  at  8  j^  a  lb.  will 
cost  56  ^  ? 

16.  In  questions  10-15  the  number  that  you  found 
in  each  case  was  the  number  of  what  ?  This  number 
is  often  called  the  ratio  of  the  quantity  to  the  unit. 


LESSON  1 


3 


17.  How  many  widths  of  carpet  3  ft.  wide  are 
needed  to  carpet  a  room  6  yd.  wide  ?  In  terms  of 
what  unit  must  you  express  both  the  quantity  6  yd. 
and  the  unit  3  ft.  before  dividing  ? 

18.  How  many  strips  of  paper  2  ft.  wide  are 
needed  to  cover  the  walls  of  a  room  whose  perimeter 
is  20  yd.? 

19.  How  many  quarts  of  milk  at  6i^  a  qt.  will 
cost  1 1.80? 

20.  In  the  following  examples  what  numbers 
express  the  measurements  ? 

Quantity  Unit  Quantity 

$45  15  2  1b. 

20  da.         5  da.  10  qt.  1  pt. 

i  hr.  6  min.  4  yd.  2  ft. 


14 
2bu. 


2  dimes 
8  qt. 


Unit 
4  OZ. 
3pt. 

2  ft. 
3  gal.  3  qt.  5  qt. 


16  sq. 


2-; 


in.  square 


21.  What  is  the  ratio  of  each  quantity  in  exam- 
ple 20  to  its  own  unit?  What  is  the  ratio  of  the 
unit  to  the  quantity  ? 

22.  A  man  earns  $15  a  week  and  spends  $11. 
In  how  many  weeks  will  he  save  $  136  ?  What  is 
the  unit  here  ? 

23.  Two  boats  travel  up  a  river  at  the  rate  of  12 
and  9  mi.  per  hour,  respectively.  If  they  start 
together,  when  will  the  faster  be  36  mi.  ahead  ? 


4  MENTAL  ARITHMETIC 

24.  A  farmer's  wife  sold  10  lb.  of  butter  at  24  ^  a 
lb.  and  with  the  proceeds  bought  sugar  at  5  ^  a  lb. 
How  many  pounds  did  she  buy  ?  What  is  the  quantity 
here  ? 

25.  A  man  travels  12  hr.  at  the  rate  of  4  mi.  an 
hr.  and  returns  to  the  starting-point  at  the  rate  of 
6  mi.  an  hr.     How  long  does  it  take  him  to  return  ? 

Lesson  2 

1.  Make  a  mental  picture  of  a  line  18  in.  long 
divided  into  3  equal  parts.  How  long  is  the  unit 
of  measure  that  divides  it  into  3  parts  ? 

2.  What  unit  would  divide  it  into  6  parts? 
2  parts  ?     9  parts  ?    ' 

3.  Find  the  units  that  divide  each  of  these 
quantities  into  4  parts  :  20  in. ;  1  ft.  4  in. ;  $  24 ; 
1  bu. ;   1  bu.  8  qt. ;   2  doz.   oranges. 

4.  In  question  3  how  did  you  find  the  value 
of  the  unit  in  each  case? 

5.  How  would  you  find  the  size  of  the  unit  divid- 
ing a  quantity  into  4  parts  ?  5  parts  ?  6  parts  ? 
Into  any  number  of  parts? 

6.  In  questions  7-13  name  the  quantity  and  the 
number  in  each  case.  How  will  you  find  the  value 
of  each  unit  ? 

7.  If  3  lb.  of  coffee  cost  96^,  what  is  the  price 
of  1  lb.  ? 


LESSON  2  5 

8.  A  freight  train  runs  72  mi.  in  4  hr.     What 
is  the  average  speed  per  hr.  ? 

9.  If  2  gal.  of  milk  is  worth  48^,  what  is  1  qt. 
worth  ? 

10.  If  the  cost  of  fencing  one  side  of  a  field  8  rd. 
long  is  $44,  what  is  the  cost  per  rd.?     Per  yd.? 

11.  Nine  yards  of  cloth  cost  $2.16  ;  find  the  price 
per  yd. 

12.  I  sold  10  cd.  of  wood  at  $6  a  cord,  and  with 
the  proceeds  bought  12  bbl.  of  flour.  What  was  the 
cost  per  bbl.? 

13.  A  rectangle  8  in.  long  and  9  in.  wide  is  cut 
into  6  equal  rectangles.     What  is  the  area  of  each  ? 

14.  In  the  following  examples  find  the  units  of 
measurement : 


Quantity 

Number 

Quantity 

Number 

24  h. 

6 

8x25^ 

40 

4x9^ 

12 

$20 

3  +  1 

80  X  $8 

40 

1  lb.  6  oz. 

11 

300  bu. 

10 

66  yd. 

4 

11.08 

9 

3bu. 

6 

15.  State  what  each  number  in  example  14  is  the 
ratio  of.  In  example  14  what  is  ^,  j^,  etc.,  the  ratio 
of? 

16.  How  long  is  the  line  that  is  measured  5  times 
by  the  unit  6  in.? 


6  MENTAL  ARITHMETIC 

17.  I  fill  a  pail  by  emptying  into  it  a  2-qt.  measure 
full  of  water  6  times.  How  much  water  does  the 
pail  hold  ? 

18.  In  examples  16  and  17  how  did  you  find  the 
whole  quantity?  When  you  are  given  the  number 
and  unit  how  do  you  find  the  quantity  ? 

19.  In  examples  20-23  name  the  number  and  the 
unit  in  each  case. 

20.  Find  the  cost  of  7  lb.  of  oatmeal  at  4  ^  a  lb. 

21.  A  farmer  sold  4  horses  at  $80  apiece.  What 
did  he  receive  for  them? 

.^  22.  Two  men  separate,  one  walking  west  at  the 
rate  of  3  mi.  an  hour,  the  other  east  at  5  mi.  an 
hr.     How  far  apart  will  they  be  in  9  hr.  ? 

23.  Two  boys  ride  along  the  same  road  in  the 
same  direction.  The  first  rides  9  mi.  an  hr.  and  the 
second  7  mi.  How  far  apart  will  they  be  in  3  hr. 
after  the  faster  passes  the  slower  ? 

24.  In  the  following  examples  find  the  quantities 
measured  by  the  numbers  and  the  units  : 


Number 

Unit 

Number 

Unit 

6 

4  sq.  ft. 

6 

$.20 

1* 

8^ 

6 

3  for  5^ 

3 

42  mi. 

2doz. 

3j^  apiece 

6 

3  tens 

8 

2 

mi.  +  3  mi. 

5  tens 

3 

7 

9 

mi. —6  mi, 

25.    Quantity  -^  Unit  =  ?    Quantity  -j-  Number  =  ? 
Number  x  Unit  =  ? 


ADDITION 

Lesson  3 

1.  A  line  8  in.  long  is  divided   into   two  parts. 
What  can  the  lengths  of  these  parts  be  ? 

2.  A  line  is  made  up  of   parts  4  in.,  5  in.,  and 
6  in.  long.     What  is  the  length  of  the  line  ? 

3.  Eight   is   how  much  less  than  10?     5  is  how 
much  greater  than  2  ?     Why  is  8  +  5  =  10  +  3? 

4.  8  +  6  =  10  +  ?  =  ?     7  +  9  =  10  +  ?  =  ? 

5.  As  in  question  4  add  the  following  : 

5        64        6        7        5        36        4        5 

6677778888 


7 

8 

2 

9 

4 

8 

6 

7 

5 

3 

8 

8 

9 

9 

9 

9 

9 

9 

9 

9 

6. 

Memorize  the 

sum  of 

2 

8 

4 

5 

3 

4 

5 

6 

4 

6 

9 

8 

7 

6 

9 

8 

7 

6 

9 

8 

6 

6 

6 

7 

6 

7 

7 

8 

8 

7 

7 

9 

8 

7 

9 

8 

9 

8 

9 

6 

8  MENTAL  AKITHMETIC 

The  Statement  of  Balance  for  example  7  is: 
The  sum  spent  =  5^  +  9^. 

Give  the  statement  of  balance  for  each  of  the  following 
examples  of  this  lesson  as  you  solve  them. 

7.  Miriam  paid  5^  for  street  car  fare  and  spent 
9^  for  a  yard  of  ribbon.  What  did  she  spend 
altogether  ? 

8.  A  man  spent  $8  for  a  coat  and  $4  for  a  vest. 
What  did  both  cost  him  ? 

9.  Ethel  spent  6^  for  apples  and  9j^  for  oranges. 
How  much  did  she  spend  in  all  ? 

10.  A  lady  paid  1 7  for  a  lounge  and  $5  for  a 
chair.     What  did  both  cost  ? 

11.  A  grocer  buys  pineapples  at  9^  apiece  and 
sells  them  at  a  gain  of  4^  each.  Find  the  selling 
price. 

12.  Add  4  to  6,  9,  3,  7,  8,  5,  6,  4,  7,  6,  9. 

13.  Add  7  to  8,  6,  3,  5,  4,  8,  9,  7,  2,  5,  9. 

14.  Add  5  to  2,  8,  4,  9,  8,  7,  5,  9,  6,  8,  7. 

15.  Add  9  to  7,  4,  9,  6,  3,  9,  8,  5,  2,  4,  1. 

16.  Add  3  to  5,  7,  9,  6,  4,  5,  7,  3,  6,  9,  8. 

17.  Add  8  to  5,  9,  7,  5,  6,  2,  8,  6,  9,  7,  4. 

18.  Add  6  to  9,  5,  8,  7,  4,  8,  5,  7,  4,  9,  6. 

Statement  of  Balance 
The  weight  =  4  times  2  lb.  +  3  lb. 

19.  What  is  the  weight  of  4  2-lb.  cans  of  corn 
and  3  lb.  of  cheese  ? 


LESSON  4  9 

20.  What  is  the  cost  of  2  spools  of  thread  at  4^ 
apiece  and  a  paper  of  needles  at  5^^? 

21.  What  must  I  pay  for  a  paper  of  pins  at  7)^  and 
2  bunches  of  tape,  at  4  ^  apiece  ? 

22.  Add  5  to  each  of  these  numbers. 

(a)  6,  26,  66,  73,  36,  96,  87,  16,  49,  56, 
{b)  3,  32,  13,  63,  43,  88,  53,  24,  73,  93. 
(0  8,  68,  46,  28,  58,  38,  77,  18,  86,  98. 
((^)  4,  54,  14,  44,  69,  84,  94,  73,  29,  34. 
(^)  7,  17,  38,  57,  24,  87,  65,  97,  46,  77. 
(/)  5,  25,  34,  65,  15,  96,  15,  45,  57,  89. 
(^)  9,  49,  19,  28,  34,  59,  99,  79,  88,  69. 
(A)  If  to  any  number  ending  in  9  you  add  5,  in 
what  figure  does  the  sum  end  ? 

23.  Add  in  turn  each  of  the  following  numbers  to 
the  numbers  given  in  question  22  : 

2,  6,  3,  7,  1,  9,  4,  10,  8. 

24.  Saturday  Helen  had  58  ^  in  the  Penny  Savings 
and  Monday  she  put  in  7^  more.  How  much  had 
she  then  ? 

25.  I  paid  $  36  for  a  gold  watch,  f  9  for  a  chain, 
and  $  5  for  a  gold  ring.     Find  the  total  cost. 

Lesson  4 

Give  the  statement  of  balance  in  each  example. 
1.    I  paid  98^  for  a  Dewey  sailor  suit  and  8^  for 
a  flag.     Find  the  cost  of  both. 


8 

4 

6 

9 

7 

8 

5 

9 

8 

4 

8 

9 

47 

63 

75 

87 

76 

94 

8 

9 

4 

6 

6 

2 

6 

7 

6 

7 

8 

9 

72 

37 

63 

88 

43 

94 

10  MENTAL  ARITHMETIC 

2.  Find  the  cost  of  2  yd.  of  ribbon  at  12^  a  yd. 
and  1  yd.  of  cambric  at  9  ^. 

3.  A  man  earns  f  15  a  week,  the  older  of  his  two 
sons  $7,  and  the  younger  1 6.  Find  their  total 
weekly  earnings. 

4.  Add  the  following  : 


5. 


6.  Find  the  cost  of  8  bunches  of  firecrackers  at 
4  ^  apiece  and  5  cents'  worth  of  punk. 

7.  I  walked  3  mi.  to  the  railway  station,  trav- 
elled 58  mi.  by  train,  and  rode  6  mi.  from  the  station 
to  my  destination.     Find  the  total  distance. 

8.  What  is  the  sum  of  9  and  6  ?  19  and  6  ? 
6  and  29  ?  16  and  9  ?  9  and  38  ?  6  and  69  ?  86 
and  9  ?     9  and  56  ? 

9.  What  is  the  sum  of  49,  4,  8  ?  5,  62,  4  ?  6, 
48,7?     3,58,9?     7,25,6?     6,78,8?     9,78,6? 

10.  Peoria  is  55  mi.  north  of  Springfield.  How 
far  apart  are  two  places,  one  6  mi.  north  of  Peoria 
and  the  other  8  mi.  south  of  Springfield  ? 

11.  I  paid  5  j^  for  a  basket,  59  j^  for  a  tub,  and  10^ 
for  a  tin  cup.     Find  the  total  cost. 


LESSON  4  11 

12.  A  milkman  delivers  milk  to  five  families  that 
take  3  qt.  a  day,  and  to  four  families  taking  2  qt.  a 
day.  How  much  milk  do  these  nine  families  buy 
each  day  ? 

13.  A  boy  riding  his  bicycle  at  the  rate  of  8  mi. 
an  hour  goes  from  A  to  B  in  2  hr.,  from  B  to  C  in 

3  hr.,  and  from  C  to  D  in  2  hr.     How  far  is  it  from 
A  to  D? 

14.  I  paid  5  ten-dollar  bills  and  1  five-dollar  bill 
for  a  rug.     What  did  it  cost  ? 

15.  I  bought  an  organ,  giving  for  it  6  ten-dollar 
bills,  3  one-dollar  bills,  and  half  a  dollar.  What  did 
it  cost  ? 

16.  A  lady  gave  for  a  piano  4  fifty-dollar  bills, 

4  ten-dollar  bills,  and  a  five-dollar  bill.     What  did  it 
cost  her  ? 

17.  For  my  hat  I  paid  a  two-dollar  bill,  3  dimes, 
and  a  nickel.     What  did  the  hat  cost  ? 

18.  A   lady  preserved   4   2-qt.    jars   of   cherries, 

5  2-qt.  jars  of  pears,  and  3  2-qt.  jars  of  peaches. 
How  many  quarts  of  fruit  did  she  preserve  ? 

19.  What  quantity  is  represented  by  the  sum  of 
3,  2,  4,  the  unit  of  measure  being  1 1  ?  $  10  ?  •$  100  ? 
2  mi.?   2  A.? 

20.  What  distance  is  represented  by  4  units  of 
100  mi.  each,  3  units  of  10  mi.,  and  2  units  of  1  mi.  ? 

21.  James  gave  6  dimes  and  a  nickel  for  a  history 
and  8^  for  a  block  of  paper.     Find  what  all  cost. 


12  MENTAL  ARITHMETIC 

22.  Richard  had  9  marbles,  Harry  43,  and  Thomas 
7.     How  many  had  all  three  ? 

23.  A  farmer  sold  2  horses  for  $  100  each,  a  cow 
for  $20,  and  a  sheep  for  $5.  What  did  he  get  for 
all? 

24.  A  thermometer  registers  8°  below  zero.  This 
is  how  many  degrees  below  the  freezing  point  ? 

25.  I  paid  59^  for  a  walking  stick,  the  price  of 
which  had  been  reduced  25^.  Find  the  original 
price. 

Lesson  5 

1.  Find  the  sum  of  the  following : 

32  and  20,         64  and  30,         86  and  30, 
75  and  60,         97  and  50,        58  and  80. 

2.  Find  the  sum  of  58  and  34. 
Note.      58  and  30  =  88 ;   88  and  4  =  92. 

As  in  example  2,  find  the  sum  of  the  following : 

3.  43  and  25,         26  and  73,         41  and  16, 
57  and  32,         68  and  32,         38  and  56. 

4.  37  and  65,         68  and  29,         43  and  39, 

84  and  49,  94  and  48,  55  and  77. 

5.  69  and  74,  85  and  58,  39  and  96, 
77  and  26,  28  and  93,  42  and  89. 

6.  55  and  36,  78  and  57,  96  and  91, 

85  and  69,         43  and  99,         88  and  33. 

7.  A  man  paid  $67  for  a  bicycle  for  himself  and 
1 25  for  one  for  his  son.     Find  the  cost  of  both. 


LESSON  ^  13 

8.  A  farmer  who  owned  76  A.  of  land  bought  an 
adjoining  piece  containing  24  A.  Find  the  size  of 
his  farm  after  the  .purchase. 

9.  A  field  is  66  rd.  long  and  48  rd.  wide.  How 
far  does  a  person  go  in  walking  from  one  corner  of 
the  field  to  the  opposite  corner  if  he  walks  along  the 
edge  of  the  field? 

10.  If  there  are  39  pupils  in  the  first  grade  and  44 
in  the  second,  how  many  pupils  are  there  in  the  two 
grades  ? 

11.  A  man  lost  $23  by  selling  his  horse  for  I  98. 
Find  what  the  horse  cost  him. 

12.  A  cow  is  tied  to  a  stake  with  a  rope  44  ft. 
long.  A  second  cow  is  tied  with  a  rope  7  yd.  longer. 
Find  the  length  of  the  second  rope. 

13.  A  merchant  bought  cloth  at  98  j^  a  yd.  and 
sold  it  at  a  gain  of  25^  a  yd.     Find  the  selling  price. 

14.  A  milkman  had  two  cans,  one  holding  8  gal. 
and  the  other  26  qt.  Find  the  total  capacity  of  both 
cans. 

15.  Winifred  gave  $.96  for  a  geography  and  ^  as 
much  for  a  reader.     What  did  both  cost  ? 

16.  After  taking  2  doz.  lemons  out  of  a  box  there 
remained  19  in  the  box.  How  many  were  in  the 
box  at  first  ? 

17.  A  farmer  sold  a  cow  for  $35  and  7  lambs  at 
$4  each.     What  was  the  total  selling  price  ? 


14  MENTAL  ARITHMETIC 

18.  One  winter  morning  the  thermometer  in  the 
schoolroom  registered  49°.  In  an  hour  the  tempera- 
ture increased  19°.  What  did  the  thermometer  then 
register  ? 

19.  What  is  the  distance  from  Chicago  to  Mil- 
waukee, if,  after  travelling  57  mi.  from  Chicago,  I 
still  have  23  mi.  further  to  go  to  reach  Milwaukee  ? 

20.  A  rope  was  cut  into  two  pieces.  If  the 
shorter  piece  was  5  yd.  1  ft.  and  the  longer  45  ft., 
how  many  feet  long  was  the  rope  ? 

21.  How  many  quarts  of  oats  are  there  in  a  bin 
containing  2  bu.  28  qt.?     How  many  pounds? 

22.  A  field  is  42  rd.  long  and  28  rd.  wide.  How 
many  rods  of  fencing  are  required  to  enclose  it  ? 

23.  A  bicycle  dealer  who  had  88  bicycles  in  stock 
received  a  shipment  of  4  doz.  more.  How  many  did 
he  then  have  in  stock  ? 

24.  Find  the  number  of  hours  in  2  da.  17  hr. 
Find  the  number  of  sheets  of  paper  in  2  quires  12 
sheets. 

25.  I  paid  25  units  of  $100  each  for  a  lot  and 
built  a  house  on  it  at  a  cost  of  36  units  of  $  100  each. 
Find  the  total  cost. 

Lesson  6 

1.    Find  the  sum  of  28  and  34 ;   of  62  and  45. 
Find  the  sum  of  48  and  27  ;  of  75  and  89. 


LESSON  6  15 

2.  Find  the  sum  of  56  and  17  ;  73  and  68 ;  66, 
17,  68.  How  do  you  find  the  sum  of  three  numbers 
like  56,  17,  68  ?     Find  the  sum  of  45,  24,  53. 

3.  Find  the  sum  of  the  following  numbers  :  120, 
32  ;  140,  53  ;  134,  65  ;  128,  76  ;  184,  78  ;  256,  89. 

4.  Find  the  sum  of  :  25,  32,  69  ;  48,  36,  63  ;  77, 
56,  99 ;  91,  84,  32 ;  67,  97,  83 ;  54,  69,  27. 

5.  A  lady  pays  $17  for  a  bedstead,  $61  for  a 
dresser,  and  $  21  for  a  washstand.  Find  the  entire 
cost. 

6.  A  dealer  bought  three  carloads  of  coal.  The 
first  weighed  24  T.,  the  second  22  T.,  and  the  third 
19  T.     Find  the  total  amount. 

7.  The  sides  of  a  triangular  field  are  respectively 
36  rd.,  45  rd.,  and  53  rd.     Find  its  perimeter. 

8.  Monday  a  grocer  sold  goods  to  the  value  of 
$56,  Tuesday  to  the  value  of  $62,  and  Wednesday 
to  the  value  of  $47.  Find  his  sales  for  the  first  half 
of  the  week. 

9.  A  horse  cost  $  125  and  the  harness  $  16.  What 
was  the  cost  of  both  ? 

10.  An  arithmetic  cost  S5f^,  a  history  27  j^  more. 
Find  the  cost  of  the  history.     Of  both. 

11.  A  grocer  paid  $  47  for  sugar  and  $  28  more  than 
that  amount  for  flour.     What  did  both  cost  him  ? 

12.  A  man  began  business  with  38  thousand  dol- 
lars and  in  ten  years  cleared  58  thousand  dollars. 
How  much  was  he  then  worth  ? 


IQ  MENTAL   ARITHMETIC 

13.  How  many  days  are  there  in  the  three  spring 
months  ?  In  the  three  summer  months  ?  In  the 
three  winter  months  ? 

14.  A  fruit  dealer  sold  6  oranges  for  15  ^,  9  others 
for  24  ^,  and  a  dozen  of  another  kind  for  45  ^.  How 
many  oranges  did  he  sell  and  what  did  he  get  for 
them  ? 

15.  State  how  you  solved  examples  5-14.  State 
how  you  will  solve  the  remaining  examples  in  this 
lesson. 

16.  A  young  man  earns  $7  a  week,  his  older 
brother  $4  more,  and  his  father  $6  more  than  the 
older  brother.     What  do  all  three  earn  a  week  ? 

17.  A  man  paid  $  52  for  a  buggy  and  ^  as  much 
more  for  a  horse.     Find  the  cost  of  both. 

18.  If  I  paid  25  hundred  dollars  for  one  lot,  18 
hundred  dollars  for  another,  and  22  hundred  dollars 
for  a  third  lot,  what  was  the  entire  cost  ? 

19.  How  many  nickels  are  there  in  three  piles  of 
which  the  first  contains  23,  the  second  16,  and  the 
third  11.     What  is  their  value  in  dollars  and  cents  ? 

20.  If  it  was  15  min.  after  9  o'clock  17  min.  ago, 
what  time  will  it  be  in  24  min.? 

21.  A  young  man  entered  the  navy  16  yr.  ago  at 
the  age  of  25.    How  old  will  he  be  12  yr.  from  now  ? 

22.  A  farmer  divided  his  farm  among  his  three 
sons,  giving  to  the  eldest  85  A.,  to  the  second  son 
76  A.,  and  to  the  youngest  68  A.  What  was  the 
size  of  his  farm  ? 


I 


LESSON  6  17 

23.  A  merchant  bought  clothes  at  $  15  a  suit  and 
overcoats  at  f  12  apiece,  and  sold  them  at  an  advance 
of  $  5  each.  What  must  a  customer  pay  for  a  suit 
of  clothes  and  an  overcoat  ? 

24.  A  dealer  bought  lemons  at  25^  a  dozen,  and 
oranges  at  35  ^  a  dozen.  If  he  sold  them  so  as  to 
gain  10)2^  a  dozen,  what  did  he  receive  for  a  dozen 
of  each  ? 

25.  Three  children  buy  a  hammock.  The  first 
contributes  75  ^,  the  second  69  ^,  and  the  third  41  ^, 
What  did  the  hammock  cost  ? 


SUBTRACTION 
Lesson  7 

Statement  of  Balance 
Ex.  1.   The  other  part  =  9  in.  —  5  in. 
Ex.  2.   The  quantity  left  =:  3  gal.  -  3  qt. 
Give  the  statement  of  balance  for  each  example  in  this 
lesson  as  you  work  it. 

1.  A  string  9  in.  long  is  cut  into  two  parts  one 
of  which  is  5  in.  long.     How  long  is  the  other  part  ? 

2.  From  a  can  containing  3  gal.  of  milk  3  qt.  are 
poured  out.     How  many  quarts  are  left  in  the  can  ? 

3.  If  2  gal.  of  milk  are  poured  into  an  empty  can 
that  will  hold  15  qt.,  how  many  quarts  are  needed 
to  fill  it  ? 

4.  In  examples  2  and  3  the  quantities  to  be  sub- 
tracted are  not  expressed  in  terms  of  the  same  unit. 
Explain  this  statement.  What  must  be  done  with 
quantities  expressed  in  terms  of  different  units  be- 
fore you  can  find  their  difference  ? 

5.  A  boy  has  8  nickels  and  buys  6  oranges  at 
a  nickel  each.     Find  how  many  cents  he  has  left. 

6.  Two  boys  were  playing  a  game  of  marbles. 
If  one  of  them  got  7  out  of  15,  how  many  did  the 
other  get  ? 

18 


LESSON  7  19 

7.  A  piece  2  ft.  3  in.  long  is  cut  from  a  rope  6  ft. 
11  in.  long.     Find  the  length  of  the  other  piece. 

8.  If  in  paying  for  a  pair  of  shoes  that  cost  $6 
I  gave  the  clerk  2  five-dollar  bills,  what  change 
should  I  receive  back  ? 

9.  A  lady  has  7  2-qt.  jars  of  fruit.  If  her  family 
eats  1  qt.  a  da.  for  6  da.,  how  many  jars  of  fruit  will 
she  have  at  the  end  of  that  time  ? 

10.  If  the  perimeter  of  an  oblong  is  18  in.  and 
the  length  5  in.,  what  is  the  width  ? 

11.  Two  vessels  going  in  opposite  directions  are 
17  mi.  apart  one  hour  after  they  pass  each  other. 
If  one  is  going  8  mi.  an  hour,  what  is  the  rate  of 
the  other  ? 

12.  A  post  12  ft.  long  is  4  ft.  below  ground. 
How  long  is  the  part  above  ground  ? 

13.  From  a  piece  of  cloth  5  yd.  long  a  piece  6  ft. 
long  is  cut.     How  many  feet  of  cloth  are  left  ? 

14.  I  paid  2  dimes  for  a  collar  and  got  back  a 
nickel  in  change.     What  did  the  collar  cost  me  ? 

15.  I  paid  16  f^  for  a  pound  of  butter,  giving  three 
different  coins  in  payment.     What  were  they  ? 

16.  A  merchant  buys  berries  for  5^  a  box  and 
sells  them  at  the  rate  of  2  for  15^.  Find  his  gain 
on  two  boxes.     On  a  crate  of  16  boxes. 

17.  A  boy  picked  8  qt.  of  berries  in  two  days. 
If  he  picked  3  qt.  1  pt.  the  first  day,  how  much  did 
he  pick  the  second  day  ? 


20  MENTAL  ARITHMETIC 

18.  From  a  line  that  is  6  times  the  measuring  unit 
I  cut  off  a  part  that  is  4  times  as  long  as  the  measur- 
ing unit.  How  many  inches  long  is  the  remainder, 
the  unit  being  3  in.? 

19.  A  horse  which  travels  at  the  rate  of  8  mi.  an 
hour  goes  from  A  to  B  in  3  hr.,  and  from  B  to  C  in 
5  hr.  How  much  farther  is  it  from  B  to  C  than 
from  A  to  B  ? 

20.  What  quantity  is  the  difference  between  11 
and  7,  the  unit  of  measure  being  2  in.?  |2  ?  3  ft.? 
4qt.?     110? 

21.  How  much  greater  are  2  units  of  f  5  than  3 
units  of  $2.  What  is  the  ratio  of  this  difference 
to  $  2  ?     Of  $2  to  this  difference  ? 

22.  A  piece  of  cloth  3  ft.  long  and  2  ft.  wide  is 
cut  from  a  square  yard  of  cloth.  How  many  square 
feet  in  the  remainder  ? 

23.  I  ordered  half  a  pound  of  cheese  and  received 
10  oz.  This  was  how  many  ounces  more  than  were 
ordered  ? 

24.  If  a  man  sleeps  7  hr.  a  day,  he  sleeps  how 
much  less  than  half  the  time  ? 

Lesson  8 

1.    Subtract  4  from  each  of  the  following  : 
(a)    6,  16,  26,  38,  46,  56,  69,  76,  81,  96. 
(5)    13,  33,  12,  63,  43,  88,  53,  24,  73,  93. 
((?)    8,  68,  46,  28,  58,  38,  77,  18,  86,  98. 


LESSON  8  21 

((^)   4,  54,  14,  44,  63,  84,  94,  72,  24,  34. 

(0    T,  17,  38,  57,  23,  87,  61,  97,  46,  77. 

(/)  10,  20,  33,  60,  74,  90,  41,  80,  72,  50. 

(^)   5,  25,  34,  65,  75,  96,  15,  45,  57,  89. 

(K)    11,  51,  63,  21,  31,  41,  93,  82,  61,  71. 

(0    9,  49,  19,  28,  34,  59,  99,  79,  88,  69. 

(m)  12,  32,  64,  92,  82,  41,  49,  52,  72,  22. 

(n)  If  from  any  number  ending  in  2  you  subtract 
4,  in  what  figure  does  the  difference  end  ?  Apply 
a  similar  question  to  each  part  of  example  1. 

2.  Subtract  in  turn,  where  possible,  each  of  the 
following  numbers  from  those  given  in  example  1 : 

2,     6,     3,     7,     1,     9,     5,     10,     8. 

3.  Harry  takes  a  step  34  in.  long  and  John  5  in. 
shorter.     How  far  does  John  step  ? 

4.  Robert  can  jump  6  ft.  high  and  James  7  in. 
less.     How  high  can  James  jump  ? 

5.  I  took  21  min.  to  walk  to  the  depot  and  then 
found  that  I  was  5  min.  late  for  the  train.  How 
long  before  train  time  did  I  leave  home  ? 

6.  Two  boys  run  a  race  of  100  yd.  When  the 
first  completes  the  course  the  second  is  7  yd.  behind. 
How  far  did  he  run  ? 

7.  If  it  is  8  min.  after  9  o'clock,  how  long  before 
it  will  be  10  o'clock. 

8.  A  man  paid  125  for  an  overcoat  and  a  pair 
of  shoes.  If  the  shoes  cost  $6,  what  did  the  over- 
coat cost  ? 


22  MENTAL   ARITHMETIC 

9.  A  lady  buys  a  quire  of  paper  and  then  writes 
7  letters,  using  a  sheet  a  letter.  How  much  paper 
has  ^he  left  ? 

10.  Out  of  a  class  of  43  pupils  9  are  absent.  How 
many  are  present  ? 

11.  A  thermometer  that  registered  72°  fell  5°. 
What  did  it  then  register  ? 

12.  A  grocer  sold  a  peck  of  beans,  taking  them 
out  of  a  bushel  basket.  If  the  basket  was  full  of 
beans,  how  many  quarts  were  left  in  it  ? 

13.  Subtract  2  from  each  of  these  numbers  : 
(a)  8,  13,  7,  18,  16,  9,  23,  11,  5,  14,  18. 

(^)  27,  46,  19,  38,  51,  84,  17,  20,  67,  42,  59. 

((?)  10,  37,  18,  63,  24,  12,  26,  15,  74,  18,  30. 

((^)  15,  18,  34,  21,  45,  82,  83,  36,  29,  13,  36. 

(g)  121,  132,  105,  143,  137,  183,  149,  158. 

14.  Subtract  in  turn,  where  possible,  each  of  the 
following  numbers  from  those  given  in  example  13  : 

3,     6,     1,     7,    4,     9,     8,     10,     5. 

15.  A  man  earns  $54  a  month,  and  saves  |9  a 
month.     How  much  does  he  spend  ? 

16.  From  12  tens  subtract  8  tens.  From  26  hun- 
dred subtract  9  hundred. 

17.  Out  of  4  doz.  eggs  half  a  dozen  were  found 
to  be  broken.     How  many  eggs  were  whole  ? 

18.  Subtract  30  from  each  of  these  numbers  : 

45,     67,    81,    93,    76,    124,    159,    116,    142,    138. 


LESSON  9  23 

19.  Subtract  in  turn,  where  possible,  each  of  the 
following  numbers  from  those  given  in  example  18  : 

20,     60,     40,     70,     10,     90,     80,     50,     100. 

20.  Subtract  28  from  76. 
Note.     76  -  20  =  56 ;  56  -  8  =  48. 

21.  Subtract  34  from  56  ;  23  from  49 ;  42  from 
73  ;  15  from  68  ;     52  from  95  ;  63  from  99. 

22.  There  are  45  pupils  in  a  class  of  whom  21  are 
boys.     How  many  girls  are  there  ? 

23.  If  the  freezing-point  is  32°,  how  much  above 
the  freezing-point  is  the  temperature  when  the  ther- 
mometer registers  67°  ? 

24.  A  man  is  59  yr.  of  age  and  his  son  23  yr. 
How  much  older  is  the  father  than  the  son  ? 

25.  In  a  race  between  two  bicyclists  the  faster 
has  gone  59  mi.  and  the  slower  47  mi.  How  far 
is  he  behind  ? 

Lesson  9 

.1.  Subtract  48  from  92  ;  68  from  84 ;  37  from  78  ; 
58  from  96  ;  32  from  61 ;  54  from  82 ;  33  from  71 ; 
46  from  65. 

2.  Find  the  difference  between  45  and  80  ;  67  and 
91 ;  23  and  56 ;  16  and  95 ;  16  and  82 ;  43  and  88 ; 
54  and  29  ;  70  and  22. 

3.  Subtract  each  of  the  following  numbers  from 
100: 

62,     37,     24,     81,     98,     56,     73,     45,     19. 


24  MENTAL  ARITHMETIC 

4.  Subtract  each  of  these  quantities  from  $1.20: 
87^,  69^,  43^,  25^,  72^,  5Qj^,  92^,  18^. 

5.  Find  the  difference  in  cents  on  subtracting 
48^  from  11.25;  76^  from  11.50  ;  89^  from  $1.67  ; 
$1.25  from  $1.74  ;  $3.48  from  $5. 

6.  The  following  bargains  were  advertised  in  a 
Chicago  paper.  Find  the  reduction  in  price  in  each 
case. 

1.  Pillow  covers  12^,  reduced  from  40)^. 

2.  Pillows  23^,  reduced  from  75^. 
8.    Silk  89^,  reduced  from  $1.50. 

4.  Silk  crepons  $1.50,  reduced  from  $2.25. 

5.  Wool  checks  $1.38,  reduced  from  $3.50. 
.    6.    Silk  waists  $1.98,  reduced  from  $4. 

7.  Umbrellas  59^',  reduced  from  $1.25. 

8.  Parasols  89^,  reduced  from  $2.50. 

9.  Bicycle  skirts,  $2.75,  reduced  from  $5. 

10.  Bicycle  suits,  $2.48,  reduced  from  $5. 

11.  Fancy  ribbon  46^,  reduced  from  $1. 

12.  Black  silks  62^,  reduced  from  $1.25. 

13.  Silkaline  18  ^,  reduced  from  50  ^. 

14.  Portieres  $2.25,  reduced  from  $6. 

15.  Lace  curtains  80^,  reduced  from  $2. 

7.  A  drover  bought  60  sheep  from  three  farmers. 
If  he  bought  18  from  the  first  and  16  from  the  sec- 
ond, how  many  did  he  buy  from  the  third  ? 

8.  A  man  who  has  $  75  in  the  bank  deposits  $  36 
more,  and  shortly  after  withdraws  $48.  How  much 
still  remains  in  the  bank  ? 


LESSON  9  25 

9.  A  and  B  start  from  two  places  99  mi.  apart  and 
travel  toward  each  other.  The  first  day  A  goes  38 
mi.  and  B  25  mi.  How  far  are  they  then  apart  ? 
How  far  is  A  from  B's  starting-place  ?  How  far  is 
B  from  A's  starting-place  ? 

10.  A  and  B  start  from  two  places  85  mi.  apart 
and  travel  in  the  same  direction.  The  first  day  A 
travels  48  mi.  and  B  27  mi.  How  far  are  they  then 
apart  ?  How  far  is  A  from  B's  starting-place  ?  How 
far  is  B  from  A's  starting-place  ?  Show  that  there 
are  two  solutions  to  this  example. 

11.  Mr.  Stevens  paid  $68  for  one  horse  and  $75 
for  another,  and  sold  them  both  for  $  160.  Find  his 
gain. 

12.  A  farmer  sold  two  bags  of  apples  for  $1.50. 
With  this  money  he  bought  a  hoe  for  32^,  a  rake  for 
45^,  and  a  spade  with  the  remainder.  What  did  the 
spade  cost? 

13.  A  merchant  sells  cloth  at  $1.25.  He  lowers 
the  price  39^,  thus  losing  12  j^  a  yd.  Find  what  the 
cloth  cost  him  a  yard. 

14.  A  boy  shot  an  arrow  34  yd.  up  the  road  and 
another  down  the  road  45  yd.  His  brother  brought 
them  to  him.     How  far  did  he  have  to  walk  ? 

15.  A  merchant  sells  silk  at  $2.50  a  yd.  He 
lowers  the  price  75 j^  and  then  sells  at  ISj^  above 
cost.     Find  what  the  silk  cost  him  a  yard. 


MULTIPLICATION 

Lesson  10 

1.  Review  the  multiplication  table,  page  134. 

2.  What  unit  of  area  is  used  to  measure  the  area 
of  an  oblong  9  in.  by  6  in.  ?  Of  a  table  3  ft.  by  2  ft.  ? 
Of  the  floor  of  a  room  6  yd.  by  5  yd.  ?  Of  a  farm  ? 
Of  a  section  of  land  6  mi.  long  and  4  mi.  wide  ? 

3.  Find  the  area  of  each  oblong  given  in  example  2. 

4.  Find  the  area  of  the  floor  of  each  of  the  rooms 
whose  dimensions  are,  respectively :  5  yd.,  4  yd. ;  8 
yd.,  7  yd.  ;  11  yd.,  9  yd. ;  12  yd.,  10  yd. 

5.  How  do  you  find  the  number  of  units  of  area 
in  an  oblong  ? 

6.  A  garden  bed  6  ft.  long  and  4  ft.  wide  is  sur- 
rounded by  a  walk  1  ft.  wide.  What  is  the  entire 
area  of  the  bed  and  walk  ? 

7.  A  flower  garden  is  surrounded  by  a  walk  2  ft. 
wide.  The  length  of  the  walk  and  garden  is  12  ft. 
and  their  width  10  ft.     Find  the  area  of  the  garden. 

8.  Find  the  number  of  square  inches  in  each  of  the 
stamped  linen  squares  whose  sides  are,  respectively, 
5,  7,  9,  and  12  in. 

26 


LESSON  10  27 

9.  A  rug  occupies  the  centre  of  a  square  room  and 
its  sides  are  everywhere  2  ft.  from  the  wall.  If  the 
length  of  the  rug  is  8  ft.,  what  is  the  area  of  the 
room  ? 

10.  How  long  is  a  township  ?  How  wide  ?  What 
is  its  area  ? 

11.  What  is  the  difference  in  area  between  an  8-in. 
square  and  8  sq.  in.  ?  Between  a  square  whose  side 
contains  6  units  of  length  and  an  oblong  containing 

6  units  of  area  ? 

12.  A  township  contains  a  forest  3  mi.  long  and  2 
mi.  wide,  the  rest  being  divided  into  farms.  How 
many  square  miles  of  farm  land  are  there  in  the 
township  ? 

13.  A  grass  plot  10  ft.  square  has  in  the  centre  a 
square  flower  bed  whose  side  is  4  ft.  Find  how 
many  square  feet  of  grass  are  in  the  plot. 

14.  In  a  room  14  ft.  long  and  12  ft.  wide  is  a  rug 
whose  sides  are  everywhere  2  ft.  from  the  wall.  Find 
its  area. 

Reduce  to  lower  denominations: 

15.  8  yd.  2  ft.;  4  yd.  1  ft.;  12  yd.  2  ft.;  9  yd. 
2  ft. ;  13  yd.  1  ft. ;  20  yd.  2  ft.  How  do  you  reduce 
yards  and  feet  to  feet  ? 

16.  6  ft.  3  in. ;  10  ft.  8  in. ;  4  ft.  6  in. ;   9  ft.  2  in. ; 

7  ft.  11  in. ;   12  ft.  6  in.     How  do  you  reduce  feet 
and  inches  to  inches  ? 


28  MENTAL   ARITHMETIC 

17.  4  sq.  yd.  6  sq.  ft.;  6  sq.  yd.  7  sq.  ft.;  12  sq, 
yd.  4  sq.  ft. ;  9  sq.  yd.  5  sq.  ft. ;  7  sq.  yd.  8  sq.  ft. ; 
12  sq.  yd.  3  sq.  ft.  How  do  you  reduce  square 
yards  and  square  feet  to  square  feet? 

18.  8qt.  Ipt.;  12qt.  Ipt.;  7gal.  3qt.;  9  gal. 
2  qt. ;    4  gal.  6  pt. ;    12  gal.  4  pt. 

19.  6  bu.  3  pk. ;  10  bu.  2  pk. ;  9  pk.  1  qt. ;  11  pk. 
2  qt.;    13  bu.  Ipk.;    8  pk.  7  qt. 

20.  4  wk.  3  da. ;  8  wk.  5  da. ;  12  wk.  4  da. ;  5  hr. 
40  min. ;    4  hr.  18  min. ;    7  min.  35  sec. 

21.  5  dimes  6^;  9  dimes  3^;  12  dimes  5^;  6  m. 
4  dm. ;    8  m.  7  dm. ;    11  m.  8  dm. 

22.  Find  the  perimeter  of  the  rooms  whose  dimen- 
sions are  respectively:  5  yd.,  4  yd.;  8  yd.,  6  yd.; 
12  yd.,  8  yd.;    22  ft.,  18  ft. 

23.  How  do  you  find  the  perimeter  of  a  room  ? 
Find  the  perimeter  of  a  room  8  yd.  long  and  7  yd. 
wide.  If  this  room  is  3  yd.  high,  find  the  area  of  its 
four  walls. 

24.  Find  the  area  of  the  four  walls  of  a  room  18  ft. 
long,  12  ft.  wide,  and  9  ft.  high.  How  do  you  find 
the  area  of  the  four  walls  of  a  room  ? 

25.  How  many  square  feet  of  wall  will  a  roll  of 
wall  paper  8  yd.  long  and  18  in.  wide  cover? 

26.  If  the  width  of  a  farm  is  measured  by  4  units 
and  its  length  by  6  units,  how  many  units  in  its 
perimeter? 


LESSON  11  29 

Lesson  11 

In  examples  1  and  2  the  statements  of  balance  are: 
The  weekly  profit  =  6  x  |15. 

The  quantity  of  provisions  =  the  rations  of  6  men  for  12  da. 
As  you  solve  the  examples  of  this  lesson,  give  the  statements 
of  balance. 

1.  A  merchant  averages  a  profit  of  $15  a  day. 
How  much  is  that  a  week  ? 

2.  A  quantity  of  provisions  will  last  6  men  12  da. 
How  long  will  it  last  1  man  ? 

3.  If  8  men  can  do  a  piece  of  work  in  9  da.,  how 
long  will  it  take  2  men  to  do  it  ? 

4.  If  7  men  can  do  a  piece  of  work  in  4  da.,  how 
many  men  can  do  it  in  2  da.  ? 

5.  If  two  men  start  from  the  same  place  and  travel 
in  opposite  directions,  one  at  the  rate  of  4  mi.  an  hr. 
and  the  other  at  the  rate  of  5  mi.  an  hr.,  how  far 
apart  will  they  be  in  3  hr.? 

6.  How  far  apart  will  the  men  in  example  5  be  in 
8  hr.  if  they  travel  in  the  same  direction  ? 

7.  Two  men  start  60  mi.  apart  and  travel  toward 
each  other  at  the  rate  of  3  mi.  an  hr.,  the  other  at 
the  rate  of  4  mi.  an  hr.  How  far  apart  will  they  be 
in  6  hr.? 

8.  How  far  apart  will  the  men  in  example  7  be  in 
6  hr.  if  they  travel  in  the  same  direction  ?  Show 
that  there  are  two  solutions. 


80  MENTAL  ARITHMETIC 

9.  I  bought  7  T.  of  coal  at  $6  a  ton,  and  handed 
in  payment  5  ten-dollar  bills.  How  much  change 
should  I  receive  ? 

10.  For  how  much  must  I  sell  6  bicycles  that  cost 
$28  each  to  gain  $45?  Statement  of  balance:  the 
selling  price  =  the  cost  +  the  gain. 

11.  A  farmer  sold  8  sheep  at  16  each  and  5  lambs 
at  i  3  each.     How  much  did  he  get  for  them  all  ? 

12.  A  tailor  has  a  piece  of  cloth  containing  25  yd. 
If  he  should  use  13  yd.  and  sell  the  remainder  at  $5 
a  yd.,  what  would  he  receive  for  it  ? 

13.  A  and  B,  travelling  toward  each  other,  met  in 
12  hr.  How  far  apart  were  their  starting-points  if 
A  travelled  9  mi.  an  hr.  and  B  8  ? 

14.  I  bought  38  cows  for  $  35  each,  and  sold  them 
for  143  each.     Find  my  profit. 

15.  A  man  bought  a  horse  for  $  90,  paid  $  3  a  week 
for  his  expenses,  and  received  |5  a  week  for  his 
work.  At  the  end  of  8  wk.  he  sold  him  for  $80. 
Did  he  gain  or  lose  by  the  whole  transaction,  and 
how  much  ? 

16.  If  a  man  travels  from  his  home  10  hr.  at  6  mi. 
an  hr.,  and  then  returns  at  the  rate  of  7  mi.  an  hr. 
for  5  hr.,  how  far  is  he  then  from  home  ? 

17.  What  is  the  cost  of  a  5-lb.  pail  of  butter  at 
19  ^  a  lb.,  and  2  lb.  of  coffee  at  35  ^  a  lb.? 


LESSON   12  31 

18.  A  man  drives  4  mi.  to  the  station,  rides  on 
the  train  for  3  hr.  at  33  mi.  an  hr.,  and  then  drives 
from  the  depot  into  the  country  a  distance  of  8  mi. 
What  is  the  entire  distance  ? 

19.  A  crate  of  raspberries  holds  24  pt.  boxes. 
How  many  quarts  is  this  and  what  is  a  crate  worth 
at  6  j^  a  pint  box  ? 

20.  What  is  the  cost  of  2  crates  of  currants  at  90  ^ 
a  crate,  2  crates  of  black  raspberries  at  75  ^  a  crate, 
and  a  crate  of  red  raspberries  at  a  $  1.05  ? 

21.  How  long  will  it  take  a  lady  to  look  over  2 
crates  of  black  raspberries  containing  16  qt.  boxes 
each,  if  she  can  do  2  boxes  in  9  min.? 

22.  A  grocer  buys  soap  at  $2.15  per  box  of  60 
bars  and  sells  it  at  5  i^  a  bar.  Find  his  gain  on  a 
dozen  boxes. 

23.  What  will  4  rows  of  wire  fencing  cost  for  a 
lot  100  ft.  long  and  50  ft.  wide  at  6  j^  a  yard  ? 

24.  A  certain  map  is  drawn  so  that  4  mi.  is  rep- 
resented by  1  in.  On  this  map  a  certain  district  is 
a  square  whose  side  is  2  in.     What  is  its  area  ? 

Lesson  12 

1.  In  terms  of  what  unit  is  the  volume  of  a  prism 
4  in.  long,  3  in.  wide,  and  2  in.  thick  expressed? 
What  is  its  volume  ? 


32  MENTAL  ARITHMETIC 

2.    Find  the  volumes  of  prisms  of  the  following 
dimensions : 


Length 

Width 

Thickness 

Length 

Width 

Thickness 

6  in. 

5  in. 

4  in. 

8  ft. 

4  ft. 

4  ft. 

8  in. 

6  in. 

4  in. 

10  ft. 

8  ft. 

7  ft. 

Tin. 

5  in. 

Sin. 

4  yd. 

3  yd. 

2  yd. 

How  many  cubic  inches  in  a  brick  8  in.  by  4  in.  by 
2  in.? 

3.  A  gallon  of  water  will  just  fill  a  tin  box  11  in. 
long,  7  in.  wide,  and  3  in.  deep.  How  many  cubic 
inches  in  a  gallon  ? 

4.  A  cubic  yard  of  earth  is  a  wagon  load.  How 
many  loads  of  earth  will  it  take  to  make  an  excava- 
tion 18  ft.  long,  12  ft.  wide,  and  6  ft.  deep  ? 

5.  How  many  cubic  inches  in  a  2-in.  cube?  3-in.? 
4-in.?    5-in.?     G-in.?     7-in.?    8-in.?     9-in.?     10-in.? 

6.  A  cubic  foot  of  water  weighs  1000  oz.  A  tank 
6  ft.  long,  2  ft.  wide,  and  2  ft.  deep  contains  how 
many  ounces? 

7.  The  distance  between  two  points,  measured 
with  a  yardstick  1  in.  too  long,  is  found  to  be  6  yd. 
What  is  the  actual  distance? 

8.  The  length  of  a  room,  measured  with  a  yard- 
stick 1  in.  too  short,  is  found  to  be  8  yd.  What  is 
the  actual  length  ? 

9.  Name  two  factors  of  15,  18,  24,  28.  Multiply 
36  by  15,  using  the  two  factors  of  15  as  multipliers. 
Multiply  similarly  36  by  18,  24,  and  28  respectively. 


LESSON  12  33 

10.  Why  is  it  correct  in  order  to  multiply  a  num- 
ber by  5  to  annex  zero  and  divide  by  2  ? 

By  this  method  multiply  the  following  numbers 

by  5: 

36,    45,     78,     96,     124,     256. 

11.  Find  the  cost  of  cementing  the  floor  of  a  cellar 
21  ft.  long  and  18  ft.  wide  at  15  j^  a  sq.  yd. 

12.  Find  the  cost  of  digging  a  cellar  5  yd.  long, 
4  yd.  wide,  and  2  yd.  deep  at  24  i^  a  cu.  yd. 

13.  If  3  ft.  is  the  unit  of  length,  what  is  the  unit 
of  area?  If  the  measuring  unit  is  a  line  3  in.  long, 
find  the  area  of  an  oblong  whose  dimensions  are  6 
and  4  tiines  the  measuring  unit. 

14.  What  is  the  cost  of  2  lb.  8  oz.  of  butter  at  16  ^ 
alb.? 

15.  An  electric  car  makes  15  trips  a  day,  carry- 
ing on  the  average  20  passengers  a  trip  at  5^  a  fare. 
How  much  did  it  earn  in  1  mo.  of  30  da.? 

16.  Why  is  it  correct,  in  order  to  multiply  a  num- 
ber by  25,  to  annex  two  zeros  and  divide  by  4  ? 

By  this  method  multiply  the  following  numbers 
by  25: 

24,    18,     35,     27,     25,    19. 

17.  I  bought  19  chickens  at  25^  apiece.  What 
change  should  I  get  out  of  a  five-dollar  bill  ? 

18.  A  merchant  bought  10  bu.  of  peaches  at  $1.75 
a  bu.,  and  1  bu.  being  spoiled,  he  sold  the  remainder 
at  $  2  a  bu.     Find  his  gain  or  loss. 


34  MENTAL  ARITHMETIC 

19.  A  lady  bought  a  remnant  of  silkaline  contain- 
ing 3|-  yd.,  and  a  remnant  of  ribbon  containing  5^ 
yd.  of  ribbon.  The  price  of  each  was  12  j^  a  yd.,  find 
the  cost. 

20.  From  a  piece  of  cloth  containing  32  yd.  there 
were  sold  at  different  times  11  yd.,  8  yd.,  and  9  yd. 
The  remainder  was  put  on  the  remnant  counter  to 
sell  at  25  ^  a  yd.     Find  the  selling  price. 

21.  A  lady  gave  her  little  girl  a  dollar  bill  with 
which  to  buy  a  2-lb.  roll  of  butter  at  22^  a  lb.,  and 
1  lb.  of  coffee  at  S5j^  a  lb.  What  change  did  she 
bring  home  to  her  mother  if  she  was  allowed  to  buy 
5  ^  worth  of  candy  ? 

22.  A  city  store  advertises  7  ^  muslin  at  3  ^  a  yd. 
and  25  ^  lace  curtains  at  10  j^  a  yd.  A  lady  buys  4  yd. 
of  the  latter  and  6  yd.  of  the  former.  How  much 
less  does  she  pay  than  if  she  had  bought  them  at  the 
regular  price  ? 

23.  If  6  bbl.  of  flour  cost  $33,  what  will  24  bbl. 
cost  at  the  same  rate  ? 

24.  If  1  pt.  of  maple  syrup  costs  12  i^,  what  will 
1  gal.  cost  at  the  same  rate  ? 

25.  How  many  feet  in  1  rod  ?  2  rods  ?  20  rods  ? 
Wire  fencing  is  sold  at  14.90  per  bale  of  20  rd., 
or  at  2  ^  per  ft.  How  much  is  saved  by  buying  20 
rd.  by  the  bale  instead  of  by  the  foot  ? 


LESSON  13  35 

Lesson  13 


B  F  C 

1.  The  triangle  a  is  what  part  of  the  oblong  AFBJD  ? 
The  triangle  h  is  what  part  of  the  oblong  AFOU? 
The  triangle  ABO  (i.e.  a  +  b^  is  what  part  of  the 
oblong  BBOU? 

2.  To  find  the  area  of  a  triangle  multiply  one-half 
the  measure  of  its  base  by  that  of  its  altitude.  Memo- 
rize this  rule. 

3.  If  ABC  represents  a  triangle  whose  base  BCis 
8  in.  and  altitude  AF  is  4  in.,  what  is  the  area  of  the 
triangle  represented?     Draw  the  triangle. 

4.  Find  the  areas  of  the  following  triangles  : 
Base  Altitude  Base  Altitude 

10  in.  6  in.         2  ft.  1  ft.  8  in. 

16  in.  12  in.         2  ft.  4  in.  1  ft.  10  in. 

3  yd.  1  ft.  4  yd. 

4  yd.  2  ft.  5  yd.  1  ft. 

3  yd.  10  in. 

4  rd.  8  yd. 

5.  What  is  the  area  of  a  triangular  garden  bed 
whose  base  is  2  yd.  and  altitude  1  yd.  2  ft.? 


18  ft. 

10  ft. 

24  ft. 

22  ft. 

36  yd. 

20  yd. 

80  yd. 

60  yd. 

36 


MENTAL  ARITHMETIC 


6.  The  gable  end  of  a  barn  measures  20  ft.  and 
the  perpendicular  distance  from  the  ridge  of  the  barn 
to  the  base  is  11  ft.  How  many  square  feet  of  lum- 
ber does  it  contain  ? 


D  C 

7.  Which  is  greater  the  triangle  a  or  a'  ?  Which 
is  greater  the  parallelogram  ABQB  or  the  rectangle 
I]FCB1 

8.  How  do  you  find  the  area  of  a  rectangle  such 
as  EFOD  ?     Of  a  parallelogram  such  as  ABCD  ? 

9.  To  find  the  area  of  a  parallelogram  multiply  the 
measure  of  the  base  (CD)  by  the  measure  of  the  alti- 
tude (CF).     Memorize  this  rule. 

10.  Find  the  areas  of  the  following  parallelograms: 

Base  Altitude  Base  Altitude 

8  ft.  6  ft.  16  rd.  12  rd. 

12  yd.  8  yd.  24  mi.  20  mi. 

Base  =  9  units  of  length.  Altitude  =  4  units  of 
length. 

11.  Name  three  pairs  of  factors  of  20.  Give  the 
base  and  altitude  of  three  parallelograms  whose  area 
in  each  case  is  20  sq.  in. 


LESSON   13 


37 


E 

A 

B 

F 

/ 

; 

/ 

c 

\ 

^ 

\ 

/ 

\ 

H 


12.  Name  the  two  parallel  sides  of  the  trapezoid 
AB  CD,  Which  is  larger,  the  triangle  a  ov  a'  1  b  or 
b'? 

13.  If  you  add  the  triangles  a^  and  b'  to  the  large 
central  figure  c,  you  get  what  figure  ?  If  you  add 
the  triangles  a  and  b  to  the  central  figure  c,  what  fig- 
ure do  you  get?  Which  is  larger  the  trapezoid 
ABCI)  or  the  rectangle  EFQEl 

14.  Show  by  actually  measuring  that  GiR  is  one- 
half  the  sum  of  AB  and  CD,  AB  and  CD  are  called 
the  bases  of  the  trapezoid  ABCD, 

15.  To  find  the  area  of  a  trapezoid  multiply  the 
measure  of  one-half  the  sum  of  its  bases  or  parallel 
sides  by  the  measure  of  its  altitude.  Memorize  this 
rule. 


16.  The  lengths  of  the  parallel  sides  of  a  trapezoid 
are  12  ft.  and  8  ft.  and  its  altitude  is  5  ft.  Find  its 
area. 


Bases 

Altitude 

11  in.,    9  in. 

7  in. 

14  ft.,    8  ft. 

6  ft. 

12  ft.,    9  ft. 

4  ft. 

13  ft.,  10  ft. 

8  ft. 

^8  MENTAL   ARITHMETIC 

17.  Find  the  areas  of  these  trapezoids: 

Bases  Altitude 

1  ft.   7  in.,  1  ft.  3  in.     10  in. 
5  yd.  2  ft.,  2  yd.  1  ft.       3  yd. 

2  yd.  1  ft.,  1  yd.  2  ft.       2  ft. 
4  yd.  2  ft.,  3  yd.  1  ft.       6  ft. 

18.  The  border  of  a  rug  consists  of  4  equal  trape- 
zoids whose  bases  are  10  ft.  and  6  ft.  If  the  width 
of  the  border  is  2  ft.,  find  its  area. 

19.  In  each  corner  of  the  centrepiece  of  a  rug  is 
a  trapezoid  whose  parallel  sides  are  13  in.  and  9  in. 
and  whose  altitude  is  14  in.     Find  their  area. 

20.  In  the  centre  of  a  rug  is  a  hexagon  which  can 
be  divided  into  two  equal  trapezoids  whose  bases  are 
2  ft.  and  1  ft.,  and  altitude  1  ft.  4  in.  Find  its 
area. 

21.  How  do  you  find  the  area  of  a  rectangle  ? 
Triangle  ?     Parallelogram  ?     Trapezoid  ? 


DIVISION 

Lesson  14 

In  the  following  examples  give  the  statements  of 
balance  : 

1.  What  must  8  mi.  be  multiplied  by  to  give 
72  mi.?  If  a  boy  travels  72  mi.  on  a  bicycle  at  the 
rate  of  8  mi.  an  hr.,  how  many  hours  is  he  travelling? 

2.  6  is  one  factor  of  42,  what  is  the  other  ?  If  6 
lb.  of  dates  cost  42^,  what  is  the  price  per  lb.? 

3.  At  5^  a  lb.  how  many  pounds  of  sugar  can  you 
buy  for  80)^? 

4.  A  horse  trotted  48  mi.  in  6  hr.  How  many 
miles  an  hour  did  he  average  ? 

5.  In  the  preceding  four  questions  name  the  given 
quantity,  the  measuring  unit,  and  the  number  of 
units  in  the  quantity. 

6.  Reduce  to  quarts:  18  pt.;  25  pt. ;  36  pt.; 
43  pt. ;  57  pt. ;  128  pt. ;  141  pt. 

7.  Find  the  number  of  strips  of  carpet  each  2  ft. 
wide  required  to  carpet  a  room  16  ft.  wide;  20  ft. 
wide;    26  ft.  wide;    17  ft.  wide. 

8.  Reduce  to  yards:  42  ft.;  96  ft.;  105  ft.; 
141ft.;  186  ft.;  204  ft.;  167  ft.;  343  ft.;  196  ft. 

39 


40  MENTAL   ARITHMETIC 

9.  How  many  strips  of  carpet  each  3  ft.  wide 
will  be  required  to  carpet  a  room  24  ft.  wide  ? 
33  ft.  wide?      20  ft.  wide? 

10.  Reduce  to  gallons:  28  qt. ;  37  qt. ;  36  qt.; 
86  qt.;  107  qt.;  260  qt. 

11.  Find  the  number  of  school  weeks  in  20  school 
days.     In  35  da.     In  100  da.     In  200  da. 

12.  If  a  man  works  6  days  a  week,  in  how  many 
weeks  will  he  work  96  da.?  150  da.?  192  da.? 
222  da.?     264  da.? 

13.  Reduce  to  weeks:  84  da.;  98  da.;  126  da.; 
175  da. ;  240  da. ;  365  da. 

14.  Reduce  to  pecks:  40  qt.;  58  qt.;  72  qt.; 
125  qt.;  197  qt.;  215  qt.;  288  qt. 

15.  Reduce  to  square  yards  :  81  sq.  ft.;  108  sq.  ft. ; 
164  sq.  ft. ;  200  sq.  ft. ;  243  sq.  ft. 

16.  Reduce  to  dimes:  80^;  120^;  93^;  168^; 
248^;  290^;  312^;   415^. 

17.  Reduce  to  meters:  40  dm.;  70  dm.;  82  dm.; 
94  dm. ;  99  dm. :  100  dm. 

18.  How  many  yards  in  1  rod  ?  Measure  off  1  rd. 
in  the  schoolroom.  Measure  this  distance  again  with 
a  unit  half  a  yard  in  length.  Count  the  number  of 
half-yards  in  1  rd.     How  many  ? 

19.  What  is  the  number  of  rods  in  33  half -yd.? 
6Q  half -yd.?     88  half -yd.? 

20.  Divide  by  11:  75;  92;  132;  189;  210;  265; 
331 ;  484. 


LESSON  15  41 

21.  Reduce  to  feet:  72  in.;  89  in.;  117  in.; 
169  in. ;  254  in. ;  288  in. 

22.  How  many  dozen  are  there  in  60  units? 
93  units  ?     384  units  ? 

23.  State  how  you  reduce  a  quantity  expressed  in 
pints  to  quarts.  Quarts  to  gallons.  Quarts  to 
pecks.  Inches  to  feet.  Feet  to  yards.  Square 
feet  to  square  yards.  Days  to  weeks.  Cents  to 
dimes. 

Lesson  15 

1.  What  must  5  lb.  be  multiplied  by  to  get  40  lb.? 
40  lb.  of  butter  is  packed  into  pails  each  containing 
5  lb.     How  many  ? 

2.  4  is  one  factor  of  36,  what  is  the  other  ?  If 
4  yd.  of  ribbon  cost  36^,  what  is  the  price  per  yd.? 

3.  A  pail  contains  9  qt,  of  water.  How  many 
quarts  will  be  left  in  the  pail  after  8  pt.  are  taken 
out  ? 

4.  What  will  a  dozen  oranges  cost  at  the  rate 
of  3  for  lOj^? 

5.  How  many  lemons  can  I  buy  for  50  ^  at  the 
rate  of  4  for  10^? 

6.  A  man  travels  18  mi.  at  the  rate  of  6  mi.  an 
hr.,  and  returns  immediately  at  the  rate  of  9  mi. 
an  hr.     How  many  hours  is  he  gone  ? 

7.  A  man  drove  to  the  city  in  3  hr.  at  the  rate 
of  8  mi.  an  hr.,  remained  2  hr.  to  transact  business, 
and  returned  at  the  rate  of  6  mi.  an  hr.  How  long 
was  he  gone  ? 


42  MENTAL  ARITHMETIC 

8.  A  dairyman  traded  4  horses  worth  $  90  apiece 
for  cows  worth  $30  apiece.  How  many  cows  did 
he  get  ? 

9.  If  3  lb.  of  rice  cost  24  ^,  what  will  2  lb.  cost  ? 

10.  I  select  the  cheaper  of  two  rngs,  one  of  which 
costs  $48  and  the  other  $Q6,  With  the  difference 
in  price,  how  many  chairs  can  I  buy  at  1 6  apiece  ? 

11.  A  room  is  10  yd.  2  ft.  long.  How  many  steps 
will  a  boy  take  in  walking  the  length  of  the  room  if 
he  steps  2  ft.  each  time  ? 

12.  A  room  is  12  ft.  wide.  How  many  strips  of 
carpet  each  2  ft.  wide  are  needed  to  carpet  it  ?  How 
many  yards  of  carpet  are  needed  if  the  room  is  15  ft. 
long  ? 

13.  A  room  is  15  ft.  wide  and  18  ft.  long.  How 
many  yards  of  carpet  are  needed  to  carpet  it,  if  the 
carpet  is  3  ft.  wide,  the  carpet  running  lengthwise  ? 

14.  A  room  is  20  ft.  wide  and  24  ft.  long.  How 
many  yards  of  carpet  are  needed  to  carpet  it,  if  the 
carpet  is  2  ft.  wide,  the  carpet  running  lengthwise  ? 

15.  If  two  5-lb.  pails  of  butter  cost  $1.05,  what 
will  20  Ibo  cost  ? 

16.  If  6  bu.  of  corn  cost  $  3,  what  will  30  bu.  cost  ? 

17.  What  is  the  cost  of  12  boxes  of  berries  at 
3  for  25^? 

18.  A  farmer's  wife  sold  to  a  grocer  8  lb.  of  butter 
at  18^  a  lb.  and  took  in  exchange  raisins  at  12^  a  lb. 
How  many  pounds  of  raisins  did  she  get  ? 


LESSON   16  43 

19.  A  man  divided  3  lb.  8  oz.  of  candy  equally 
among  4  children.  How  many  ounces  did  each 
get? 

20.  Some  workmen  work  9  hr.  a  day  at  24^  an 
hour.  They  demand  the  same  daily  wages  for  an 
8-hr.  day.     How  much  would  that  be  per  hour  ? 

21.  What  is  the  cost  of  6  qt.  1  pt.  of  milk  at 
24^  a  gal.? 

22.  To  how  many  customers  does  a  milkman  sell 
8  gal.  1  qt.  of  milk  if  he  sells  3  pt.  to  each  customer  ? 

23.  I  bought  7  chickens  giving  for  them  a  $2  bill 
and  receiving  in  change  25^.  What  did  they  cost 
apiece  ? 

24.  A  grocer  buys  6  baskets  of  peaches  at  30.^  a 
basket.  He  uses  two  baskets  and  sells  the  remainder 
for  what  all  cost  him.  Find  the  selling  price  of  each 
basket. 

25.  Find  the  number  of  strips  of  paper  in  a  room 
20  ft.  long  and  18  ft.  wide,  the  paper  being  2  ft. 
wide. 

Lesson  16 

1.  Find  the  weight  of  an  iron  bar  5  yd.  1  ft. 
long  of  which  2  ft.  weigh  13  lb. 

2.  A  block  of  stone  is  10  in.  long,  8  in.  wide, 
and  6  in.  thick.  Find  its  weight  if  5  cu.  in.  weigh 
lib. 


44  MENTAL  ARITHMETIC 

3.  A  grocer  bought  a  crate  of  red  raspberries 
(24  pt.)  at  5^  a  pint.  He  kept  some  for  his  own 
use  and  sold  the  remainder  at  6  ^  a  pint  box  for  what 
all  cost.  How  many  boxes  did  he  keep  for  his  own 
use  ? 

4.  An  express  train  takes  5  hr.  25  min.  to  travel 
225  mi.  If  it  makes  stops  of  5  min.  each  at  3  differ- 
ent places  and  one  stop  of  10  min.,  find  the  average 
rate  at  which  the  train  goes. 

5.  A  railway  train  travels  at  the  rate  of  2  mi. 
in  3  min.     What  is  its  rate  per  hour? 

6.  If  8  yd.  of  ribbon  cost  96  ^,  what  will  3  yd. 
1  ft.  cost  at  the  same  rate  ? 

7.  What  number  measures  the  area  of  an  oblong 
containing  24  sq.  in.,  the  unit  being  1  sq.  in.  If  its 
length  is  6  in.,  what  number  measures  its  length, 
the  unit  being  1  in.  What  number  measures  its 
width  ?     What  is  its  width  ? 

8.  How  do  you  find  the  width  of  an  oblong  when 
its  area  and  length  are  given  ? 

9.  Find  the  width  of  the  oblongs  given  in  the 
first  two  columns,  and  the  length  of  those  given  in 
the  last  two  : 


Area 

Length 

Area 

Width 

30  sq.  in. 

6  in. 

70  sq.  yd. 

5  yd. 

48  sq.  ft. 

8  ft. 

264  sq.  yd. 

11yd. 

63  sq.  yd. 

9  yd. 

36  sq.  mi. 

6  mi. 

8  sq.  yd. 

12  ft. 

96  sq.  mi. 

8  mi. 

LESSON   16  45 

10.  A  strip  of  carpet  containing  6  sq.  yd.  is  2  ft. 
wide.     How  long  is  it  ? 

11.  The  area  of  the  ceiling  and  floor  of  a  room  24 
ft.  long  is  96  sq.  yd.     What  is  its  width  ? 

12.  Into  how  many  parts  each  2  in.  long  can  a  line 
8  in.  long  be  divided  ?  6  in.  long  ?  Into  how 
many  2-in.  squares  can  an  oblong  8  in.  long  and 
6  in.  wide  be  divided  ? 

13.  An  oblong  is  36  in.  long  and  24  in.  wide.  Into 
how  many  3-in.  squares  can  it  be  divided?  4-in. 
squares?  6-in.  squares?  8-in.  squares?  12-in. 
squares?    Square  feet? 

14.  How  many  tiles,  each  6  in.  square,  are  needed 
for  a  piece  of  tiling  5  ft.  long  and  1  ft.  6  in.  wide? 

15.  A  10-in.  cube  can  be  cut  into  how  many  5-in. 
cubes  ?     How  many  2-in.  cubes  ? 

16.  A  cubic  foot  can  be  cut  into  how  many  3-in. 
cubes?     How  many  4-in.  cubes? 

17.  How  many  pasteboard  boxes,  each  edge  of 
which  is  8  in.,  can  be  packed  into  a  box  3  ft.  4  in. 
long,  2  ft.  8  in.  wide,  and  2  ft.  deep,  inside  measure- 
ment? 

18.  A  farmer  got  256  bu.  of  wheat  off  a  field  con- 
taining 8  A.  What  is  the  yield  per  acre  worth  at 
90^  a  bushel? 

19.  How  long  will  it  take  a  boy  to  go  a  bicycle 
trip  of  336  mi.,  riding  8  mi.  an  hr.  for  7  hr.  a  day? 


46  MENTAL  ARITHMETIC 

20.  On  what  day  will  two  boys  finish  a  bicycle 
trip  of  648  mi.,  riding  9  mi.  an  hr.  for  6  hr.  a  day 
and  resting  on  Sunday,  if  they  start  on  Wednesday  ? 
If  they  start  on  Monday? 

21.  Two  boys  travel  in  opposite  directions  at  the 
rate  of  3  and  4  mi.  an  hr.  In  how  many  hours  after 
parting  will  they  be  56  mi.  apart? 

22.  Divide  $40  between  two  brothers  so  that  the 
older  gets  $3  for  every  |2  that  the  younger  gets. 
What  is  the  unit  that  measures  $40?  What  num- 
ber arises  from  the  measurement  ?     How  many  $3  ? 

23.  A  farmer  divided  a  farm  of  160  A.  between 
his  two  sons,  giving  the  older  5  A.  for  every  3  A. 
he  gave  the  younger.     Find  the  share  of  each. 

24.  If,  in  example  23,  the  farmer  gave  his  younger 
son  $  2000  in  order  to  give  him  as  much  as  the  older 
son,  what  was  the  farm  worth  per  acre  ? 

25.  I  gave  f  1.20  for  a  pound  of  coffee  and  a  pound 
of  tea.  If  the  tea  cost  three  times  as  much  as  the 
coffee,  find  the  cost  of  each. 

Lesson  17 

Before  solving  these  problems  let  different  pupils 
contribute  what  they  can  to  their  solution. 

1.  A  man  bought  an  equal  number  of  pigs  and 
sheep  for  $81.  Each  pig  cost  $3  and  each  sheep  $4. 
How  many  of  each  did  he  buy  ? 


LESSON  17  47 

2.  In  example  1  name  the  whole  quantity  and  the 
unit  of  measure.     Name  both  in  example  3. 

3.  A  merchant  sold  cloth  at  12  a  yd.,  and  twice 
as  much  at  $3  a  yd.,  the  whole  amounting  to  $216. 
How  many  yards  of  each  kind  did  he  sell? 

4.  A  merchant  sold  silk  at  $4  a  yd.,  and  three 
times  as  much  at  $  2  a  yd.,  the  whole  amounting  to 
$90.     How  many  yards  of  each  kind  did  he  sell? 

5.  A  grocer  mixed  5  lb.  of  tea  worth  40  ^  a  lb. 
with  3  lb.  worth  80^  a  lb.  What  is  the  mixture 
worth  a  pound  ? 

6.  The  expense  of  carpeting  a  room  was  $  45  ;  but 
if  the  breadth  had  been  3  ft.  less,  the  expense  would 
have  been  $40.     Find  the  breadth  of  the  room. 

7.  The  expense  of  carpeting  a  room  was  $14.40  ; 
but  if  20  ^  more  a  yard  had  been  paid  for  the  carpet, 
the  cost  would  have  been  $19.20.  Find  the  number 
of  yards  of  carpet. 

8.  How  much  water  must  be  mixed  with  30  gal. 
of  wine  at  $3  a  gal.  in  order  to  make  it  worth  $2 
a  gal.? 

9.  Divide  $  66  among  2  men  and  3  women,  giving 
each  man  twice  as  much  as  each  woman. 

10.  State  how  you  solved  each  of  the  above  ex- 
amples. 

11.  Find    the    wages    due    a  workman  who  has 
worked  45  hr.  at  $1.80  a  day  of  9  hr.  each. 


48  MENTAL  ARITHMETIC 

12.  If  3  T.  of  coal  are  worth  |18,  and  8  T.  are 
given  for  12  thousand  shingles,  how  much  are  the 
shingles  a  thousand? 

13.  If  20  A.  of  land  are  bought  for  $600,  how 
much  per  acre  must  they  be  sold  for  that  the  pur- 
chaser may  double  his  money  ? 

14.  I  divide  33  marbles  between  two  boys,  giving 
one  9  more  than  the  other.  How  many  will  each 
have? 

15.  Into  how  many  townships  can  a  tract  of  land 
18  mi.  square  be  divided,  a  township  being  6  mi. 
square. 

16.  A  rectangle  is  12  in.  long  and  6  in.  wide.  How 
many  times  is  it  greater  than  a  triangle  whose  base  is 
6  in.  and  altitude  4  in. 

17.  Give  three  pairs  of  factors  of  18.  Give  the 
dimensions  of  three  rectangles  of  different  shape,  the 
area  of  each  of  which  is  18  sq.  in. 

18.  Give  the  dimensions  of  as  many  rectangles  as 
you  can  whose  area  is  24  sq.  in. ;  20  sq.  in. ;  36  sq.  in.; 
48  sq.  in. ;  60  sq.  in. 


SQUARE  ROOT 

Lesson  18 

1.  What  is  the  product  of  4  and  4?  Of  6  and  6? 
Of  8  and  8? 

2.  How  many  units  of  area  in  a  square  whose 
side  contains  4  units  of  length?  6  units  of  length? 
8  units  of  length? 

3.  What  is  the  length  of  the  side  of  a  square  con- 
taining 16  units  of  area?  36  units  of  area?  64  units 
of  area? 

4.  We  say  that  the  square  of  4  is  16  (42=16), 
and  that  the  square  root  of  16  is  4  (Vl6  =  4). 
62=?     V36  =  ?     82  =  ?     V64  =  ? 

5.  Find  the  square  of  each  number  from  1  to  12 
inclusive,  and  commit  to  memory. 

6.  Whatis  the  square  root  of  25?  81?  4?  121? 
1?     9?    144?     16?     64?    49?     36?     100? 

7.  What  is  the  side  of  a  square  flower  bed  that 
contains  16  sq.  ft.  ?  The  perimeter  of  a  square  room 
containing  25  sq.  yd.? 

8.  A  man  drives  around  a  square  township  con- 
taining 36  square  miles  at  the  rate  of  8  mi.  an  hour. 
How  long  does  it  take  him  ?  Give  the  statement  of 
balance  in  this  and  the  following  questions.  (The 
number  of  hours  =  the  perimeter  h-  8  mi.) 

E  49 


50  MENTAL   ARITHMETIC 

9.  Two  boys  race  on  bicycles.  They  start  together, 
and  the  first  goes  around  a  square  block  containing 
36  sq.  mi.  at  the  rate  of  10  mi.  an  hour,  and  the 
second  around  a  square  block  containing  49  sq.  mi. 
at  the  rate  of  12  mi.  an  hour.  Who  comes  in  first, 
and  by  how  many  minutes  ? 

10.  If  two  squares  whose  sides  are  respectively 
3  in.  and  4  in.  were  cut  up  and  placed  in  the  form 
of  a  square,  what  would  be  the  length  of  its  side? 

11.  If  an  8-in.  square  were  cut  out  of  a  10-in. 
square,  how  large  a  square  could  be  made  out  of  the 
remainder  ? 

12.  Find  the  length  of  the  side  of  a  chicken  coop 
in  the  form  of  a  square  containing  13  sq.  yd.  4  sq.  ft. 

13.  Find  the  squares  of  13,  14,  15,  16,  17,  18,  19, 
20,  and  25,  and  memorize  your  results. 

14.  What  is  the  square  root  of  225?  169?  289? 
400?     900?     3600?     256?     196?     625?     361? 

15.  A  city  playground  contains  675  sq.  yd.  and 
is  3  times  as  long  as  it  is  broad.  Find  its  length 
and  breadth. 

16.  A  lady  bought  ribbon,  paying  as  many  cents 
a  yard  as  she  bought  yards.  The  whole  cost  was 
$1.96.  How  many  yards  did  she  buy,  and  at  what 
price  per  yard? 

17.  Draw  two  lines  respectively  3  in.  and  4  in. 
long  at  right  angles.  Join  the  other  end  of  these 
lines.     Measure   the  line  thus   drawn.     How  long 


LESSON  18  61 

is  it?  The  length  of  this  hypotenuse  (5  in.)  of  the 
right  triangle  can  be  found  from  3  and  4  without 
measuring.  How?  Test  your  method  in  other  right 
triangles. 

18.  To  find  the  hypotenuse  of  a  right  triangle^ 
square  the  measures  of  the  lengths  of  its  base  and 
altitiide.      Take  the  square  root  of  their  sum, 

19.  Find  the  hypotenuse  of  a  right  triangle  whose 
base  is  8  ft.  and  altitude  6  ft. 

20.  Find  the  diagonals  of  an  oblong  field  whose 
sides  are  15  rd.  and  8  rd. 

21.  What  is  the  base  of  a  right  triangle  whose 
hypotenuse  is  13  ft.  and  altitude  5  ft.? 

22.  The  top  of  a  ladder  15  ft.  long  rests  against 
the  side  of  a  building  12  ft.  from  the  ground.  How 
far  is  its  foot  from  the  wall? 

23.  A  square  rug  contains  256  sq.  ft.  If  it  is 
put  in  the  centre  of  a  square  room  6  yd.  long,  how 
far  is  its  edge  from  the  wall? 

24.  Find  the  length  of  the  longest  straight  line 
that  can  be  drawn  on  the  floor  of  a  room  20  ft.  long 
and  15  ft.  wide. 

25.  A  rug  8  ft.  by  6  ft.  lies  in  the  centre  of  a 
room  16  ft.  by  12  ft.  How  far  is  the  corner  of  the 
rug  from  the  corner  of  the  room  ? 

26.  An  oblong  containing  4  3-in.  squares  is  equal 
in  area  to  how  many  2-in.  squares?  25  4-in.  squares 
are  equal  in  area  to  how  many  5-in.  squares? 


GREATEST  COMMON  MEASURE   AND 
LEAST   COMMON  MULTIPLE 

Lesson  19 

1.  Name  all  the  units  of  length  that  will  exactly 
•measure  a  line  12  in.  long  and  also  a  line  18  in.  long. 

2.  Name  the  largest  unit  that  will  exactly  meas- 
ure both  lines. 

3.  What  is  the  greatest  common  measure  of  the 
following  quantities;  15  ft.,  25  ft.;  16  yd.,  24  yd.; 
35  mi.,  56  mi.;  36  gal.,  63  gal.;  28  qt.,  40  qt.;  77 
bu.,  56  bu.;  24  1b.,  1041b. 

4.  What  is  the  largest  measure  that  can  be  used 
to  measure  the  capacity  of  two  cans  which  contain 
28  qt.  and  36  qt.?  In  this  and  the  following  exam- 
ples give  the  statements  of  balance.  (The  largest 
measure  =  the  G.  C.  M.  of  28  qt.  and  36  qt.) 

5.  What  is  the  length  of  the  longest  board  that 
can  be  used  to  build  a  fence  around  a  garden  64  ft. 
long  and  40  ft.  wide  ? 

6.  What  is  the  largest  unit  of  money  that  can  be 
used  to  pay  each  of  two  debts,  one  of  160  and  the 
other  of  $100? 

62 


LESSON  19  53 

7.  A  farmer  has  39  bu.  of  wheat  and  45  bu.  of 
corn,  which  he  puts  into  sacks  of  equal  size  without 
mixing  the  two  kinds  of  grahi.  How  many  bushels 
must  each  sack  contain  in  order  to  be  as  large  as  pos- 
sible, and  how  many  sacks  were  there  of  each  kind? 

8.  What  is  the  greatest  common  measure  of  42 
and  54?     44  and  72?     63  and  98?     75  and  85? 

9.  Prove,  in  the  case  of  the  numbers  given  in 
example  8,  that  the  greatest  common  measure  of  two 
numbers  will  also  divide  their  sum  or  difference. 

10.  A  rectangular  piece  of  tiling  1  yd.  1  ft.  6  in. 
long  and  2  ft.  wide  is  made  of  square  tiles.  Find 
the  largest  possible  tile  that  can  be  used  and  the 
number  that  will  be  required. 

11.  On  attempting  to  divide  39  marbles  between 
several  boys  there  were  4  left  over.  If  there  had 
been  58,  there  would  have  been  3  left  over.  How 
many  boys  were  there  ? 

12.  What  is  the  shortest  distance  that  will  ex- 
actly contain  the  units  4  in.  and  5  in.?  3  in.  and 
7  in.?  4  in.  and  6  in.?  6  in.  and  8  in.?  9  in.  and 
12  in.?    10  in.  and  15  in.?    20  in.  and  30  in.? 

13.  Two  bells  toll  at  intervals  of  5  sec.  and  6  sec. 
respectively,  and  begin  to  toll  at  the  same  instant. 
When  will  they  next  toll  together  ? 

14.  What  is  the  least  number  of  acres  that  will 
admit  of  being  divided  into  farms  of  60  A.  or  80  A. 
each  ? 


54  MENTAL  ARITHMETIC 

15.  Arthur  can  ride  around  a  two-mile  block  in 
12  min.  and  Cecil  in  10  min.  If  they  start  together, 
when  will  they  be  together  again  ?  How  many  times 
will  each  go  round  the  block  in  the  time  ? 

16.  What  is  the  least  sum  of  money  with  which  I 
can  buy  sheep  at  $  6  apiece  or  cows  at  $26  ? 

17.  Find  the  least  sum  of  money  with  which  you 
,  can   buy  an   exact   number  of   a   dozen   lemons   or 

bananas  at  20^  and  15)^  a  dozen  respectively. 

18.  What  is  the  least  number  which,  when  divided 
by  6  or  7,  leaves  3  as  remainder  in  each  case  ? 

19.  Find  the  least  number  which,  when  divided 
by  4,  6,  or  8,  leaves  2  as  remainder  in  each  case. 

20.  The  fore  and  hind  wheels  of  a  carriage  are 
9  ft.  and  12  ft.  in  circumference.  Find  the  least 
number  of  revolutions  of  each  that  will  bring  them 
to  the  same  relative  position  as  at  starting. 

21.  Find  the  L.  C.  M.  of  the  denominators  of  the 
following  fractions  : 

JL    1  .      2     i  .      2.     5  .      3.    1  .      2     1.      3     _8_  .     _5__      1 
2'   3  '     3'   4  '      3'   6  '      5'   3  '      3'   9  '     Y'  2  1  '      12'   18* 

22.  What  is  the  least  number  of  parts  into  which 
the  primary  unit  1  yd.  must  be  divided  in  order  that 
both  1  yd.  and  ^  yd.  may  be  expressed  by  a  whole 
number  of  parts  ?     How  many  parts  in  each  case  ? 

23.  As  in  example  22,  give  the  least  number  of 
parts  into  which  the  primary  unit  must  be  divided 


LESSON  19  55 

and  the  number  of  parts  equal  to  the  given  quanti- 
ties in  the  following  :  |  yd.,  ^  yd. ;  |^  ft.,  ^  ft. ;  |  lb., 
lib.;    f  gal.,  i  gal.;    fbu.,  |bu.;    $i,  $|. 

24.  Solve  as  in  example  23  :  |-  da.,  ^  da.;  ^  yr., 
}  yr.;  f  wk.,  ^  wk.;  |  hr.,  |  hr.;  f  min.,  f  min.; 
I  mo.,  -I  mo. 

25.  Solve  as  in  example    23  :    J,  J,  ^ ;    2'  f '  f ' 

ill.      2     3     5.      111.      3     117 

2'  t'   5  '      3'  ?'   6  '     2'   6'  y  >     ¥'    12'  t* 


FRACTIONS 

Lesson  20 

1.  If  on  measuring  a  certain  unknown  quantity 
with  a  unit  1  ft.  in  length  I  count  5  units  in  the 
quantity,  then  I  know  that  the  quantity  is  5  ft. 
Similarly,  if  I  measure  another  quantity  with  a  unit 
2  ft.  in  length  and  count  4  units  in  the  quantity,  I 
know  that  it  is  4  (2  ft.).  If  I  measure  a  third  quan- 
tity with  a  unit  \  ft.  in  length  and  count  3  units  in 
the  quantity,  then  I  know  that  it  is  3  times  the  unit 
-J  ft.  or  I  ft.  Thus  when  a  measured  quantity  is 
denoted  by  the  expression  |  ft.,  the  measuring  unit 
is  ^  ft.  and  the  number  of  units  in  the  quantity  is  3. 

Let  actual  work  he  done  in  measuring  unknown 
quantities  with  fractional  units. 

2.  I  measured  a  quantity  with  the  unit  ^  ft.  and 
counted  5  units.     What  was  the  quantity  ? 

3.  Name  the  quantities  which,  when  measured  by 
the  following  units,  give  the  numbers  indicated : 


Unit 

Number 

iyd- 

4 

Aft- 

7 

imi. 

3 

Alb. 

11   . 

ilb. 

6 

Unit 
Ihr. 

Number 

7 

|gal. 
ibu. 

idoz. 

4 

3 

13 

9 

66 


LESSON  20  67 

4.  I  measured  the  quantity  of  milk  in  a  pitcher 
with  a  measure  containing  ^  qt.,  and  in  so  doing 
filled  it  5  times.  Find  the  quantity  of  milk  in  the 
pitcher. 

5.  I  measured  a  distance  and  found  it  equal  to 
I  ft.  What  unit  did  I  use  and  how  often  did  I 
measure  ? 

6.  In  each  of  the  following  quantities  name  the 
measuring  unit  and  give  the  number  of  units  that 
measure  the  quantity :  |  ft.;  |  yd.;  |  gal.;  ^  sq.  yd.; 
U  ^-5  A  cd-5  U  sq-  nii.;  -^^  T. 

7.  What  is  the  measuring  unit  for  the  quantity 
1^  ft.?  Into  how  many  parts  do  we  think  of  the 
primary  unit,  1  ft.,  as  being  divided  to  give  the 
measuring  unit  ^  ft.? 

8.  Name  the  primary  units  indicated  by  the  fol- 
lowing quantities  and  state  into  how  many  parts  we 
think  of  them  as  being  divided  in  order  to  give  the 
measuring  units:  |  ft.;  $^q;  x\  ^^-5  I  sq.  yd.; 
-5^  cu.  yd.;  38^  A.;  |f  yr.;  ^^  mo.  Name  the 
measuring  unit  in  each  case. 

9.  Find  i  of  $10;  1  of  9  lb.;  J  of  16  mi.;  ^  of 
19  hr.;  lof  28T.;  ^  of  37  A. 

10.  How  do  you  find  |^  of  a  quantity?  (Divide 
the  quantity  by  2.)  How  do  you  find  ^  of  a  quan- 
tity?     i?     1?     1?     j?     1?     I?     Jo?     t\?     T^? 

11.  Find  the  fractions,  given  in  example  10,  of 
these  quantities :  $30;  24  mi.;  45  yr.;  66  A.;  42^. 


68  MENTAL  ARITHMETIC 

12.  Find  I  of  $12;  |  of  20^;  |  of  25^;  f  of  24 
hr.;  f  of  28  da. 

13.  How  do  you  find  |  of  a  quantity?  (Divide 
the  quantity  by  3  and  multiply  by  2.)  How  do  you 
find  J  of  a  quantity?  f  ?  |?  ^7  How  do  you 
find  any  fraction  of  a  quantity  ? 

14.  Find  f  of  18;  f  of  49;  f  of  72;  |  of  54;  ^-^ 
of  80 ;  -j\  of  88 ;  -f^  of  48. 

15.  Can  you  find  a  fraction  of  a  quantity  by  mul- 
tiplying it  by  the  numerator  and  dividing  the  prod- 
uct by  the  denominator  ?  Find  in  two  ways  the 
value  of  f  of  112 ;  f  of  $8 ;  f  of  |18 ;  f  of  |14. 

16.  By  the  method  given  in  example  15,  find  the 
value  of  the  following :  |-  of  7 ;  |  of  11 ;  |^  of  12  ; 
f  of  10 ,  f  of  15  ;  f  of  20  ;  |  of  13  ;  ^3_  of  23. 

17.  If  I  cut  a  piece  of  string  48  in.  long  into 
halves,  what  will  be  the  length  of  each  piece  ?  If  into 
thirds?  Into  fourths?  Fifths?  Sixths?  Sevenths? 
Eighths? 

In  examples  18  to  28  give  the  statements  of  bal- 
ance. 

18.  A  boy  had  30  words  in  his  spelling  lesson  and 
spelled  correctly  |  of  them.  How  many  did  he  spell 
correctly  ? 

19.  A  man  is  32  yr.  old,  and  his  wife  |  as  old. 
What  is  her  age? 

20.  James  rode  28  mi.  on  his  bicycle  in  2|  hr., 
and  Harry  |  as  far.     How  far  did  Harry  ride  ? 


LESSON  21  59 

21.  During  the  month  of  April  a  commercial  trav- 
eller was  away  from  home  |  of  the  time.  How  many 
days  was  he  away  ? 

22.  A  merchant  bought  cloth  for  60  )^  a  yard  and 
sold  it  for  I  of  the  cost.     Find  the  selling  price. 

23.  A  man  earns  $20  a  week,  and  his  son  |  as 
much.     Find  the  sum  of  their  earnings. 

24.  A  man  whose  salary  is  $30  a  week  pays  ^  of 
it  for  rent  and  -|  of  it  to  meet  other  expenses.  How 
much  does  he  spend  a  week  and  how  much  does  he 
save? 

25.  A  man  who  earns  $32  a  week  spends  ^  of  it 
for  meat  and  ^  of  the  remainder  for  groceries.  How 
much  does  he  spend  each  week  for  meat  and  grocer-, 
ies. 

26.  A  horse  which  cost  $98  was  sold  for  |  of  the 
cost  price.     Find  the  loss  on  selling. 

27.  A  house  is  sold  for  $2400,  and  |  of  that  sum 
is  paid  down.     How  much  remains  unpaid  ? 

28.  What  actual  coins  and  how  many  of  each  kind 
are  equal,  respectively,  to  |,  -^^^  and  ^^  of  $  1  ? 

Lesson  21 

1.  Find  1  of  $12;  |of$12;  iof$12;  fof$12; 
Jof$12;  |of$12;  lof$12;  f  of$12. 

2.  Which  is  greater  |  or  f  or  |  or  f  of  $  12  ?     Of 
any  quantity  ? 


60  MENTAL  ARITHMETIC 

3       SllOW    that     2_3_1_5_6_10_12_15 

1^  =  1 0  :=  |o  by  taking  these  fractions  of  f  60. 

4.  What  is  f  of  a  quantity  equal  to  ?     How  can 
you  write  a  fraction  that  is  equal  to  1  ? 

5.  How  many  hours  are  there  in  J  da.?     J  da.? 
fda.?     I  da.? 

6.  What  is  meant  by  saying  that  i  =  |  =  -|  =  |  ? 

7     l_2_^_4_5  —  6—  _?    _    ?__?_? 
/.      2-"¥-"6—  ^—  T  —  y—  1^—  16—  ?  —  ?• 

What  is  the  relation  between  the  denominator  of 
a  fraction  and  the  numerator  in  order  that  a  fraction 
may  be  equal  to  ^  ? 

8.  How  many  inches  are  there  in  ^  yd.?     |  yd.? 
I  yd.?     -^2  y^-^     Name  ten  fractions  each  equal  to  ^. 

9.  Name  fractions  equal  to  J ;  ^  ;  ^  ;  | ;  ^  ;    J  ; 
iV'  A'  T2- 

10.  What  change  can  be  made  in  the  numerator 
and  denominator  of  such  a  fraction  as  ^  without  alter- 
ing its  value  ? 

11.  I  yd.  =  ?  in.  f  yd.  =  ?  in.  f  yd.  =  ?  in. 
^8_.  yd.  =  ?  in.  12  y(j^  _  ?  in.  II  yd.  =  ?  in.  Name 
five  fractions  each  equal  to  |. 

12.  What  change  can  be  made  in  the  numerator 
and  denominator  of  such  a  fraction  as  |-  without 
altering  its  value  ? 

13.  By  what  process  is  |  changed  to  -^^  or  1|  ? 
By  what  process  is  |  or  ||  changed  to  |  ? 


LESSON  21  61 

14.  f  da.  =  ?  hr.       |  da.  =  ?  hr.       -f^  da.  =  ?  hr 
Name  five  fractions  each  equal  to  J. 

15.  Show  that  I  =  3_  =  Ji_  =  _8^  =  ^  by  finding 
the  number  of  minutes  in  each  of  these  fractions  of 
an  hour. 

16.  Name  five  fractions  equal  to  each  of  the  fol- 
lowing :  |,  A    |,  2,  |,  i    _9.. 

17.  When  we  use  the  expression  ^  yd.,  we  think 
of  1  yd.  as  divided  into  how  many  parts  ?  When 
we  change  f  yd.  to  -^^  yd.,  we  think  of  each  of  these 
parts  as  divided  into  how  many  parts  ? 

18.  Why  can  you  multiply  both  terms  of  a  frac- 
tion by  the  same  number  without  changing  the  value 
of  the  fraction  ? 

19.  1  yd.  is  equal  to  how  many  halves  of  a  yard  ? 
How  many  thirds  of  a  yard  ?  Fourths  ?  Fifths  ? 
Sixths  ? 

20.  1  ft.  is  equal  to  how  many  fourths  of  a  foot  ? 
2  ft.  is  equal  to  how  many  thirds  of  a  yard  ? 

21.  Express  $1  as  4ths  of  a  dollar;  $5  as  4ths; 
6  yd.  as  3rds  ;  2  ft.  as  12ths  ;  8  gal.  as  4ths.  In 
each  case  name  the  measuring  unit  of  the  quantity 
expressed  as  a  fraction.  Name  this  unit  also  as  an 
actual  unit  in  common  use. 

22.  Express  8  wk.  as  7ths  of  a  week ;  6  yr.  as 
12ths  ;  10  da.  as  24ths  ;  5  min.  as  60ths  ;  9  pk.  as 
8ths  ;  11  bu.  as  4ths. 


62  MENTAL  ARITHMETIC 

23.  In  each  example  in  the  preceding  question 
name  the  direct  measuring  unit  as  an  actual  unit 
of  measure  in  common  use. 

24.  What  is  1^  of  24  equal  to  ?  If  1  yd.  is  divided 
into  24  equal  parts,  into  how  many  parts  is  |  yd. 
divided  ?     ^  yd.  =  g^  yd. 

25.  Reduce  |  lb.  to  12ths ;  f  lb.  to  IGths ;  f  qt. 
to21sts;  |da.  to36ths;  $|tol5ths;  f  hr.  to  30ths. 

26.  Express  as  fractions  with  100  as  denominator  : 

113JL2.34_1 7 9_    1215111 

2^  li'>  ^'>   5'   5'   5'  ^'   10'   10'   10'   3'   3' -6'   6'  Y'   8'   9* 

Lesson  22 

1.  Reduce  the  following  to  their  lowest  terms  : 
f  ft.;  I  yd.;  ^-^^mi.;  If  da.;  Ifhr.;  If  bu.;  If  pk. 

2.  Reduce  to  their  lowest  terms  :  -f-^^  |^|,  ||,  ^|, 
1^,  J|.  How  do  you  reduce  a  fraction  to  its  lowest 
terms  ? 

3.  Reduce  to  their  lowest  terms  : 

36.        33         42         18        14         6.3 
6  0'       51'      ^T'       87'       9  9'       8f 

4.  From  a  rope  30  yd.  long  I  cut  off  20  yd.  for 
a  clothes-line.  What  part  of  the  whole  piece  was 
cut  off  ? 

5.  Express  as  a  fraction  of  2  doz. : 

4,     6,     3,     8,     10,     9,     12,     15,     20,     30,     36. 

6.  Express  as  a  fraction  of  a  peck  : 

1  qt.,  4  qt.,  2  qt.,  6  qt.,  8  qt.,  9  qt.,  12  qt.,  14  qt. 


LESSON  22  •  63 

7.  Express  as  a  fraction  of  an  hour:   20  min., 

15  min.,  40  min.,  60  min.,  45  min.,  80  min.,  90  min. 

8.  Express  as  a  fraction  of  a  bushel : 

8  qt.,    24  qt.,    28  qt.,   16  qt.,    36  qt.,    48  qt.,   80  qt. 

9.  Express  as  a  fraction  of  2  ft.  6  in. :  10  in., 
6  in.,  9  in.,  15  in.,  20  in.,  24  in.,  30  in.,  36  in., 
45  in. 

10.  What  part  of  a  mile  is  220  yd.?  880  yd.? 
440  yd.?  176  yd.?  160  yd.?  480  yd.?  What  part 
of  an  acre  is  484  sq.  yd.?  121  sq.  yd.?  242  sq.  yd.? 
968  sq.  yd.?     605  sq.  yd.? 

11.  If  16  is  used  as  a  measure  of  $12,  what  num- 
ber expresses  the  measurement  ?  If  $  6  is  used  as 
a  measure  of  $  6,  what  is  the  number  ?  If  f  6  is  used 
as  a  measure  of  1 3,  what  is  the  number?  What 
is  the  ratio  of  1 3  to  1 6  ?     ' 

12.  If  8  lb.  is  used  as  a  measure  of  48  lb.,  what 
number   expresses  the   measurement  ?      Of   32  lb.? 

16  1b.?     8  1b.?    4  1b.?     6  1b.?     2  1b.? 

13.  Frank  had  32  marbles  and  gave  12  of  them 
to  his  brother.  What  part  of  them  did  he  give 
away  ?     What  part  did  he  keep  ? 

14.  A  dealer  sold  a  bicycle  which  cost  him  §40 
at  an  advance  of  $25.  The  gain  was  what  part 
of  the  cost  ? 

15.  Of  a  farm  of  90  A.,  40  A.  are  in  wheat,  25  A. 
in  oats,  15  A.  in  corn,  and  the  remainder  in  pasture. 
What  part  of  the  farm  is  devoted  to  each  purpose  ? 


64  MENTAL  ARITHMETIC 

16.  I  paid  12^  for  berries,  14  i^  for  peas,  and  10^ 
for  beans.  What  part  of  whole  sum  was  paid  for 
each  ? 

17.  At  64^  a  yd.,  how  much  cloth  can  I  buy  for 

48^? 

18.  At  20  ^  a  quire,  what  part  of  a  quire  of  paper 
can  I  buy  for  15  i^  ?     How  many  sheets  ? 

19.  A  man  walks  a  certain  distance  in  6  hr.  What 
part  of  it  does  he  walk  in  2  hr.?  In  3  hr.?  In  4  hr.? 
Inlhr.?     Injhr.?     In  |  hr.? 

20.  I  had  $30  and  spent  J  of  it  for  an  overcoat 
and  -J  of  it  for  a  bicycle  suit.  What  was  the  cost  of 
both  ?     This  is  what  part  of  $  30  ? 

21.  The  smaller  of  two  numbers  is  16  and  their 
sum  40.     Their  difference  is  what  part  of  their  sum  ? 

22.  On  ;J  of  a  field  containing  12  A.  I  sowed  oats  ; 
on  |-  of  the  remainder  wheat.  What  part  of  the  field 
did  I  sow  in  wheat  ? 

23.  A  pipe  will  fill  a  tank  containing  300  gal.  in 
10  hr. ;  a  second  pipe  will  fill  it  in  5  hr.  How  many 
gallons  will  enter  the  tank  in  1  hr.  if  both  pipes  are 
open?  This  is  what  part  of  the  capacity  of  the 
tank  ? 

24.  A  can  do  a  piece  of  work  measured  by  20 
units  in  4  hr.,  and  B  in  5  hr.  How  many  units  of 
work  will  they  do  in  1  hr.,  working  together?  This 
is  what  part  of  the  whole  work? 


LESSON  ^3  65 

25.  A  sum  of  money  is  divided  between  A  and  B 
so  that  A  gets  twice  as  much  as  B.  What  part  of 
the  money  does  each  get?  If  the  money  is  $63, 
what  does  each  get? 

26.  I  bought  2  bu.  of  plums  put  up  in  4  boxes  of 
equal  size,  each  of  which  was  divided  into  two  equal 
parts.  What  part  of  a  bushel  did  each  part  contain? 
What  part  of  half  a  bushel  ? 

Lesson  23 

1.  1240  was  divided  between  A  and  B,  so  that  B 
received  three  times  as  much  money  as  A.  How 
many  units  represent  the  entire  sum?  A  receives 
what  part  ?    B  what  part  ?    How  much  in  each  case  ? 

2.  $180  was  divided  among  A,  B,  and  C.  B  re- 
ceived twice  as  much  as  A,  and  C  as  much  as  A  and  B 
together.  What  part  of  the  money  did  each  receive  ? 
How  many  dollars  ? 

3.  In  one  corner  of  an  oblong  8  in.  by  3  in.  is 
drawn  an  oblong  3  in.  by  2  in.  The  small  oblong 
occupies  what  part  of  the  area  of  the  large  one? 

4.  At  a  recent  election  150  Azotes  were  cast  in 
favor  of  the  successful  candidate  and  120  for  his 
opponent.  The  majority  was  what  part  of  the 
whole  number  of  votes? 

5.  A  3-in.  square  is  what  part  of  a  6-in.  square? 

6.  A  3-in.  cube  is  what  part  of  a  6-in.  cube  ? 


66  MENTAL  ARITHMETIC 

7.  A  bookcase,  6  ft.  high  and  4  ft.  wide,  is  put 
at  one  end  of  a  room  12  ft.  wide  and  9  ft.  high. 
What  part  of  the  wall  space  does  it  occupy? 

8.  A  rug  consists  of  a  centre  piece  4  ft.  square 
and  a  border  2  ft.  wide.  What  part  of  the  whole 
rug  is  the  centre  piece  ?     The  border  is  what  part  ? 

9.  A  tablecloth  10  ft.  long  is  folded  under  2  ft. 
The  part  folded  under  is  what  fraction  of  what  is 
needed  for  a  tablecloth  ? 

10.  How  often  will  a  line  6  in.  long  measure  a 
line  18  in.  long  ?  3  is  often  called  the  ratio  of  18  in. 
to  6  in. 

11.  What  part  of  a  line  18  in.  is  equal  to  a  line 
6  in.  long?  J  is  often  called  the  ratio  of  6  in.  to 
18  in. 

12.  What  is  the  ratio  of  20  in.  to  5  in.?  Of  5  in. 
to  20  in.? 

13.  What  is  the  ratio  of  6  A.  to  24  A.?  If  24  A. 
of  land  cost  1 720,  what  will  6  A.  cost  at  the  same 
rate? 

14.  If  20  bu.  of  wheat  cost  |16,  what  will  4  bu. 
cost  at  the  same  rate  ? 

15.  The  ratio  of  the  value  of  a  purse  to  the  money 
in  it  is  ^,  If  there  is  $12  in  the  purse,  what  is  the 
value  of  the  purse  ? 

16.  Give  the  multiplication  table  of  4.  Name 
numbers  of  which  4  is  the  ratio.  Of  which  ^  is 
the  ratio. 


LESSON  23  67 

17.  Give  numbers  of  which  the  ratio  is  5,  | ;  6,  1 ; 

7    1  .    1  .    1 •    JL •    JL  .    JL 

''  T '    8  '     9  '     10  '     11  '    12* 

18.  What  is  the  ratio  of  the  cost  of  5  yd.  of  rib- 
bon to  that  of  15  yd.?  If  15  yd.  of  ribbon  cost 
$1.83,  what  will  5  yd.  cost  at  the  same  rate? 

19.  What  is  the  ratio  of  3  in.  to  4  in.?  Of  4  in. 
to  3  in.?     Of  3  in.  to  8  in.?     Of  8  in.  to  3  in.? 

20.  What  is  the  ratio  of  a  line  3  2-in.  long  to  a 
line  4  2-in.  long?  Of  a  line  3  3-in.  long  to  a  line 
4  3-in.  long?  Of  a  line  made  up  of  3  units  of  any 
length  to  a  line  made  up  of  4  units  of  the  same  kind? 
Of  4  units  to  3  units  ? 

21.  What  is  the  longest  unit  that  will  exactly 
measure  lines  12  in.  long  and  20  in.  long?  How 
many  times  in  each  case?  What  is  the  ratio  of 
12  in.  to  20  in.?     Of  20  in.  to  12  in.? 

22.  What  is  the  ratio  of  12  ft.  to  20  ft.?     Of  |12 

to  120?     Of  $20  to  112?     Of  20  da.  to  12  da.?     Of 
12  da.  to  20  da.? 

23.  If  a  boy  earns  $30  in  20  da.,  what  part  of  $30 
will  he  earn  in  12  da.?     How  much  ? 

24.  If  12  yd.  of  cloth  cost  $21,  what  fraction  of 
$21  will  20  yd.  cost?     How  much? 

What  is  the  largest  unit  that  will  exactly  measure 
the  following  quantities,  and  what  is  the  ratio  in 
each  case? 


68  MENTAL  ARITHMETIC 

25.  18bu.,  27bu.;  27bu.,18bu.;  24bu.,  32bu.; 
20  bu.,  30  bu. ;  the  cost  of  30  bu.  of  potatoes  to  that 
of  42  bu. ;  of  42  bu.  to  that  of  30  bu. 

26.  If  42  bu.  of  potatoes  cost  128,  what  will  30 
bu.  cost  at  the  same  rate  ? 

27.  $28,  $35;  $48,  $72;  $45,  $25;  $25,  $45; 
$54,  $42  ;  of  the  number  of  days  it  will  take  to  earn 
$81  to  the  number  it  will  take  to  earn  ! 


28.  If  a  man  earns  $99  in  66  da.,  in  how  many- 
days  will  he  earn  $81? 

29.  24  yd.,  30  yd.;  8  qt.,  4  gal.;  6  qt.,  2  pk.; 
2  sq.  yd.,  20  sq.  ft.;  36  sheets,  2  quires. 

30.  Find  the  gear  of  each  of  the  following  28-in. 
wheel  bicycles,  in  which  the  number  of  teeth  in  the 
front  and  rear  sprockets  is  given : 

Front  sprocket  =  24,  17,  20,  25,  19,  22,  27. 
Rear    sprocket  =    8,     7,     8,     9,     8,     8,     9. 

Lesson  24 

1.  What  is  the  ratio  of  the  cost  of  6  qt.  of  berries 
to  that  of  2  qt.  ?  At  2  qt.  for  15  ^,  what  will  6  qt.  of 
blackberries  cost  ? 

2.  What  is  the  ratio  of  24  lb.  to  36  lb.?  Of  36  lb. 
to  24  lb.  ?  Which  ratio  do  you  use  to  solve  the  fol- 
lowing question:  If  24  lb.  of  coffee  cost  $8,  what 
will  36  lb.  cost  ? 


LESSON  24  69 

3.  What  is  the  ratio  of  the  length  of  the  shadow 
cast  by  a  boy  4  ft.  tall  to  that  cast  by  a  man  6  ft. 
tall?     A  man  6  ft.  tall  casts  a  shadow  9  ft.  long. 

.  How  long  is  the  shadow  cast  by  a  boy  4  ft.  tall  ? 

4.  Will  it  take  2  men  longer  or  shorter  to  do  a 
piece  of  work  than  8  men  ?  In  what  ratio  ?  If  2  men 
can  do  a  piece  of  work  in  20  da.,  in  how  many  days 
can  8  men  do  it  ? 

5.  Will  it  take  more  or  fewer  rolls  of  wall  paper 
18  in.  wide  to  cover  the  walls  of  a  room  than  of 
paper  22  in.  wide?  In  what  ratio?  If  9  rolls  of 
wall  paper  22  in.  wide  are  needed  to  paper  a  room, 
how  many  rolls  18  in.  would  have  been  needed  ? 

6.  What  is  the  ratio  of  16  oz.  to  24  oz.?  Of  24  oz. 
to  16  oz.  ?  The  carpet  on  a  certain  room  weighs  24  oz. 
to  the  yard,  and  its  entire  weight  is  45  lb.  What 
would  a  carpet  for  the  same  room  weigh  if  made  out 
of  carpet  weighing  16  oz.  to  the  lb.  ? 

7.  It  costs  $9.60  to  carpet  a  room  with  hemp  car- 
pet at  24  )^  a  yd.  What  would  it  cost  with  carpet  at 
15)^  a  yd.? 

8.  The  moulding  for  a  room  costs  $1.98  at  S^ 
a  ft.     What  would  it  cost  at  4  j^  a  ft.  ? 

9.  I  paid  1 27  for  a  hat  and  a  suit  of  clothes.  If 
the  suit  cost  8  times  as  much  as  the  hat,  find  the  cost 
of  both. 


70  ^        MENTAL   ARITHMETIC 

10.  Imagine  a  line  2^  ft.  long.  With  1  ft.  as  the 
measuring  unit,  can  you  count  the  number  of  parts  ? 
(No.)  With  ^  ft.  as  the  measuring  unit,  what  num- 
ber of  parts  can  you  count?  (9.)  Expressed  in 
terms  of  the  smaller  unit,  we  have  2^  ft.  =  |  ft. 

11.  With  the  following  quantities  and  the  given 
units,  what  number  of  parts  can  you  count  ? 

Quantity  Unit  Number 

4ift.  "       Jft.  ? 

6|yd.  lyd.  ? 

3fda.  ida.  ? 

8f  hr.  1  hr.  ? 

6f  bu.  I  bu.  ? 

Express  each  quantity  in  terms  of  the  smaller  unit. 

12.  Express  each  of  these  quantities  in  terms  of 
the  smaller  unit :  8J  mi.,  6|  A.,  4|  sq.  mi.,  3f  sq.  yd., 
12|  gal.,  16f  da. 

13.  In  reducing  8f  to  an  improper  fraction,  is  it 
more  convenient  to  think  of  8  or  of  7  as  the  multi- 
plier ?     Reduce  to  improper  fractions :  4^,  3^,  5f ,  If, 

8f ,  6f . 

14.  Reduce  the  following  to  improper  fractions : 
6f,  8f,  2|,  4f,  9f,  If,  7i,  14f,  33J,  16|,  12i  111  8^. 

15.  Among  how  many  children  can  you  divide 
$  8f  equally,  if  each  child  receives  $  ^  ? 

16.  How  many  boxes  of  candy,  each  containing 
half  a  pound,  can  be  put  up  from  S^  lb.  ? 


LESSON  24  71 

17.  How  many  weeks  will  3f  T.  of  coal  last  a 
furnace  that  burns  on  the  average  ^  T.  a  week  ? 

18.  How  often  is  5  contained  in  19  and  with  what 
remainder  ?  If  you  measure  a  line  ^-  ft.  long  with 
a  foot  rule,  how  often  could  you  measure  an  exact 
number  of  feet  ?  The  remainder  would  be  what  part 
of  a  foot  ? 

19.  The  expression  ^^-  ft.  denotes  a  quantity  made 
up  of  19  units  each  equal  to  -|  ft.  The  expression 
S^  ft.  denotes  the  same  quantity  divided  into  3  parts, 
each  equal  to  the  primary/  unit  1  ft.,  and  the  re- 
maining part  divided  into  4  parts,  each  equal  to  the 
direct  measuring  unit  of  the  fraction ;  viz.  ^  ft. 

20.  Have  you  a  clearer  idea  of  the  value  of  a  quan- 
tity when  expressed  as  ^-  ft.  or  as  3|^  ft.  ? 

21.  Express  the  following  quantities  in  terms  of 
the  primary  units  and  the  direct  measuring  units : 
f  ft.,  -2/  yd.,  ^/  mi.,  ioo  lb.,  iL9.  T.,  |f|,  $^%  {^  A., 
\^  A. 

22.  Name  the  two  units  of  measurement  in  each 
quantity  in  example  21. 

Reduce  to  integers  and  proper  fractions : 

23  18  _8_3_  9_9  18  ^Ji    _8  6 
^'^'      IV     9  '  6  '  12'  9  '  1  • 

24  JLM.  2  00.  10  0  23.    _2_5_  _5_a 
^*-    3  '   9  '  T  '  2  '  3  '  3  • 

25  111  9_5-  121  JLQ_a.  .a^ 
^^'        8  '  6  '   9  '  12  '  9  • 


72  MENTAL  ARITHMETIC 

Lesson  25 

1.  Find  the  sum  of:  2  in.  and  3  in. ;  |  ft.  and 
I  ft.;  I  pk.  and  ^  pk. ;  ^  sq.  yd.  and  |  sq.  yd.; 
I  wk.  and  |^  wk.     How  can  you  add  |  and  |^  ? 

2.  Can  you  add  ^  ft.  and  ^  ft.  as  you  added  the 
quantities  in  example  1  ?     Why  not  ? 

3.  To  what  unit  must  you  reduce  2  gal.  before 
you  can  add  it  to  3  qt.?  5  ft.  before  you  can  add  it 
to  6  in.? 

4.  To  what  must  you  reduce  J  before  you  can  add 
ittoi?     ^  +  i=?     i-i  =  ? 

5.  lof  12  =  ?  lof  12  =  ?  fof  12  =  ?  fof  12  =  ? 
i  of  12  =  ?     f  of  12  =  ? 

6.  Reduce  the  following  fractions  to  12ths  :  J,  -J, 
f ^  \i  f ^  6'  6*  How  do  you  find  the  numerator  when 
you  reduce  |  to  12ths?     (Take  |  of  12.) 

7.  Reduce  to  lOths:  |-,  |,  i  f,  |,  |. 

8.  Reduce  to  15ths :  ^  |,  f ,  |,  |,  f . 

9.  Reduce  to  IGths  :  ^,  f ,  |,  |,  |,  |. 

10.  Reduce  to  18ths :  ■^,  |,  |,  |,  ^,  ^. 

11.  Reduce  to  20ths  :  i  f ,  li  |,  If,  |,  ^-S,  ii  i_9_.. 

12.  Reduce  to  24ths  :  f ,  f ,  li  f ,  f ,  1 1,  l^,  1|,  If,  If. 

13.  Reduce  to  30ths  :  ^  li  |,  f ,  If,  f ,  ^9^,  if. 

14.  What  is  the  least  common  denominator  of  the 
fractions  f^  fi  f  ?  Reduce  them  to  equivalent  frac- 
tions having  this  least  common  denominator  for  their 
common  denominator. 


LESSON  25  73 

Reduce  to  equivalent  fractions  having  for  their 
common  denominator  the  least  common  denominator 
of  the  given  fractions  : 

15  JL    1    1    _U  .    li    2     6    A 
•^^'     2'  3'  6'   12  '    -^2'   3'  6'   9* 

16  1    -3-    2     6..    3     11    6     3 

17  1113.31       2        3 

^''     -i-)  Y^  2'  4  '     6'  6'   15'  TO' 

Ta9l1l135.513lJL6 

19.  A  piece  of  land  containing  |  A.  is  divided  into 
lots  each  containing  ^  A.  Find  the  number  of  lots 
and  their  selling  price  at  $  800  each. 

20.  Pepper  is  sold  in  bottles  each  containing  ^  lb. 
How  many  of  these  bottles  will  contain  2|-  lb.? 

21.  How  many  tins  each  containing  ^  lb.  of  mus- 
tard will  contain  all  together  2i  lb.? 

22.  A  charity  organization  divided  5  bbl.  of  flour 
among  several  families,  giving  each  family  ^  bbl. 
How  many  families  received  help  ? 

23.  I  spent  $  I  in  buying  collars  at  $  ^^  each.  How 
many  collars? 

24.  A  grocer  sold  3|-  bbl.  of  flour  in  J  bbl.  sacks 
at  i  2  each.     Find  the  total  selling  price. 

25.  A  girl  works  a  problem  in  arithmetic  on  the 
average  in  -^-^  hr.  How  many  will  she  work  at  this 
rate  in  1^  hr.? 


ADDITION  AND   SUBTRACTION  OF 
FRACTIONS 

Lesson  26 

1.  What  is  the  sum  of  ^  and  J ?  ^  and  ^?  ^  and  ^ ? 
I  and  ^?     ^  and  J? 

2.  What  is  the  sum  of  J  and  ^ ?  ^  and  ^'^  ^  and  ^ ? 
^  and  1^?     ^  and  i? 

3.  A  man  whose  lot  contained  J  A.  bought  an 
adjoining  piece  that  contained  ^  A.  How  much  did 
he  then  have  ? 

4.  A  can  do  1^  of  a  piece  of  work  in  an  hour,  and 
B  ^  of  it.  How  much  can  they  do  in  one  hour  work- 
ing together? 

5.  Find  the  difference  between  ^  and  ^ ;  J  and  ^ ; 
J  and  ^ ;  ^  and  ^ ;  ^  and  ^ ;  J  and  j^g  5  i  ^^^^  9  • 

6.  Bought  ^  bbl.  of  flour.  How  much  was  left 
after  using  ^  bbl.? 

7.  What  is  the  difference  between  J  of  my  age 
and  ^  of  it?  If  this  is  2  yr.,  how  old  am  I?  If 
6  mo.,  how  old  am  I? 

74 


LESSON  26 

8.    What  is 

the 

sum  of 

^andf? 

iandl? 

fandl? 

f  and  1  ? 

fandf? 

fandil? 

1  and  I  ? 

f  andf? 

Jandf? 

,-Vandl? 

fandJj? 

f  andl? 

75 


9.    Find  the  difference  between  the  fractions  in 
example  8. 

10.  If  ^  of  the  cost  of  a  hat  was  20  ^,  what  did  the 
hat  cost? 

11.  If  |-  of  the  value  of  a  house  is  $1200,  what  is 
J-  of  its  value?  What  is  f  of  its  value?  What  is 
the  house  worth? 

12.  If  I  of  the  size  of  a  farm  is  75  A.,  what  is  ^  of 
its  size?     How  many  acres  in  the  farm? 

13.  If  f  of  a  boy's  weight  is  45  lb.,  how  much 
does  he  weigh  ?  What  is  your  divisor  here  ?  Your 
multiplier?  With  the  fractions  f,  f,  -f-Q,  |^,  respec- 
tively, what  would  your  divisors  and  multipliers  be  ? 

14.  The  sum  of  J  and  |^  of  a  boy's  weight  is  what 
part  of  it?  If  this  is  60  lb.,  how  much  does  the  boy 
weigh  ? 

15.  A  man  invested  ^  of  his  money  in  business 
and  J  of  it  in  real  estate.  What  part  of  his  money 
did  he  invest  in  both  ?  If  this  was  1 14,000,  how 
much  money  had  he? 

16.  l-f=?        l-f=?        l-f=?       1-|=? 

X         g         .^        Q         .J.         ^2         '      ^        3'^        16* 


76  MENTAL  ARITHMETIC 

17.  A  tailor  cut  off  |  of  a  piece  of  cloth  to  make 
a  suit  of  clothes.  If  6  yd.  are  left  in  the  piece,  how 
many  did  it  contain  at  first? 

18.  A  cistern  was  ^  full  of  water,  and  during  a 
rain  filled  up  J  more.  The  water  was  then  2  ft. 
from  the  top.     How  deep  is  the  cistern? 

19.  The  trunk  of  a  tree  was  cut  into  three  parts. 
The  first  contained  ^  of  the  length,  the  second  ^,  and 
the  third  part  was  20  ft.  long.  How  long  was  the 
trunk  ? 

20.  A  merchant  sold  cloth  at  45^  a  yd.,  gaining 
J  of  the  cost.  The  selling  price  was  what  fraction 
of  the  cost  ?     What  was  the  cost  per  yard  ? 

In  examples  21-27  give  the  statements  of  balance. 

21.  A  merchant  sold  apples  at  60  ^  a  bushel,  gain- 
ing ^  of  the  cost.     Find  the  cost. 

22.  A  boy  lost  ^  of  his  marbles  and  has  30  left. 
How  many  had  he  at  first? 

23.  A  man  divides  his  lot  into  two  parts,  one  of 
which,  containing  |^  of  the  entire  lot,  he  keeps  for 
himself.  The  other  part,  which  has  a  frontage  of 
100  ft.,  he  sells.  What  is  the  frontage  of  the  entire 
lot? 

24.  Two  boys  ran  a  race,  and  when  the  winner 
had  reached  the  goal,  the  second  boy,  who  was  10  yd. 
behind,  had  run  -f^  of  the  whole  distance.  How  long 
was  the  course? 


LESSON  2f  77 

25.  Three  boys  ran  a  race,  and  when  the  winner 
reached  the  goal  the  second  boy  had  run  f  of  the 
whole  distance,  and  the  third  boy,  who  was  2  yd. 
behind  him,  had  run  |  of  the  whole  distance.  How 
long  was  the  course  ? 

26.  The  torrid  zone  extends  2^^  on  each  side  of 
the  equator.     How  many  degrees  wide  is  it? 

27.  How  many  degrees  from  the  north  pole  to  the 
equator  ?  The  frigid  zone  extends  23^-°  from  the 
north  pole,  and  the  torrid  zone  2S^°  north  from 
the  equator  ?  Find  the  number  of  degrees  in  the 
north  temperate  zone. 

Lesson  27 
1.    What  is  the  sum  of  3  and  2?     Of  l  and  J? 


Of  3 J  and  21? 

2.    Find  the  sum  of  the 

following : 

2f  and  31 

3f  and  ^ 

4|  and  2| 

4|  and  6f 

3f  and  4| 

2|  and  8^ 

2|  and  If                  4f  and  1| 
21  and  IJ         •          21  and  3f 
4f  and  7f                  |    and  2f 
How  do  you  add  two  fractions  ? 

3.    Find  the  difference  between  the  following 

2|  and  IJ 

3|  and  1^ 

3|  and  2i 
2|  and  2| 

5f  and  4f 
3|  and  1| 

78  MENTAL  ARITHMETIC 

4-  I     -I     =?  IJ-I     =? 

f     -I     =?  If-f    =? 

li_i=?  13_2_? 

6=1+?  6f-|=? 

4i-i    =?  51-1    ==? 

7|-43  =  ?  8|-4f  =  ? 

5.  How  do  you  find  the  difference  between  two 
such  fractions  as  8f  and  4|  ? 

6.  Find  the  difference  between  the  following  : 

4f    and  ^  4f    and  3|| 

12    ande^^^  6^    and  5f 

361  and  24|  31    and  2^^ 

41    and2f  12fand8| 

7.  Find  the  sum  of  the  following  : 

111  91    ,^2.    _7_ 

2'  1'  6  ^2'  %'    12 

1     2      5_  91     e^l    «3 


2'   3'   12  ^^2'  "^3'  "¥ 

»'   12'  6  ^5'  ^V  ^2 

lj_l-_l  —  ?  91_J_41—  41 

5-  +    3  ¥  -  •  ^2+  %        % 


=  ? 


*-A  +  i=?  41-2^-11    =? 

9.    A  boy  who  had  $4^  earned  |2|  more.     How 
many  dollars  did  he  then  have  ? 

10.  A  dress  was  made  from  two  remnants,  one 
containing  4|  yd.,  and  the  other  8|  yd.  How  many 
yards  did  the  dress  contain  ? 


LESSON  27  79 

11.  Two  brothers  gathered  hickory  nuts.  The 
older  gathered  2|  bu.,  and  the  younger  2^  bu.  How 
many  bushels  did  they  gather  together  ? 

12.  A  lady  is  48|  yr.  old,  and  her  husband  is  5|- 
yr.  older.     Find  his  age. 

13.  A  boy  travelled  39|  mi.  on  his  bicycle  and  the 
next  day  36|  mi.  How  much  farther  did  he  travel 
the  first  day  than  the  second  ? 

14.  What  fraction  subtracted  from  the  sum  of  ^ 
and  |-  will  leave  1  for  remainder  ? 

15.  A  farmer  divided  a  portion  of  his  farm  among 
his  three  sons,  giving  to  the  two  oldest  ^  each,  and 
to  the  youngest  ^  of  it.  What  part  of  his  farm  did 
he  divide.  If  this  was  165  A.,  how  large  was  his 
farm  ?     What  was  each  son's  share  ? 

16.  Two  boys  contributed  ^  and  ^,  respectively, 
toward  the  expenses  of  an  outing.  The  third  boy, 
who  contributed  the  remainder,  put  in  75  ^.  What 
did  the  outing  cost  and  what  did  each  contribute  ? 

17.  A  number  is  larger  by  50  than  |  of  it.  Find 
the  number. 

18.  A  man  paid  a  debt  in  four  payments.  At  first 
he  paid  |^,  then  l,  then  |  of  it,  and  finally  $500. 
Find  the  original  debt  and  the  amount  of  each  pay- 
ment. 

19.  My  furnace  burns  12 1  T.  of  coal  during  the 
winter,  and  my  neighbor's  8|  T.  How  much  more 
does  it  cost  me  for  coal  than  my  neighbor,  if  coal  is 
16  a  ton? 


Sd  MENTAL  ARITHMETIC 

20.  A  bicyclist  travelling  at  the  rate  of  8  mi.  an 
hour  goes  around  a  quadrilateral  section  of  country 
of  which  the  sides  are  6^  mi.,  5|^  mi.,  3|  mi.,  and 
6|-  mi.  How  long  is  he  gone  if  he  is  delayed  a  quar- 
ter of  an  hour  on  the  road  ? 

21.  If  a  certain  number  is  increased  by  ^  of  itself 
the  sum  will  equal  75.     Find  the  number. 

22.  At  the  rate  of  8  mi.  an  hour  how  far  will  a 
boy  ride  in  two  trips,  one  lasting  3|  hr.,  and  the 
other  4|  hr.,  if  he  stops  ^  hr.  during  each  trip? 

23.  The  difference  in  weight  between  two  parcels 
is  3|^  lb.,  and  the  lighter  parcel  weighs  12^  |  lb.  What 
is  the  weight  of  the  heavier  ?     Of  both  ? 

24.  Divide  10|  lb.  of  sugar  into  two  parcels,  one 
of  which  shall  be  1^  lb.  heavier  than  the  other. 

25.  What  fraction  must  be  added  to  the  sum  of 
J  and  ^  that  their  sum  may  be  |  ? 


I 


MULTIPLICATION   AND   DIVISION   OF 
FRACTIONS 

Lesson  28 

1.  Find  the  cost  of  10  yd.  of  cloth  at  $  |  a  yd. 
Name  the  number^  the  unit^  and  the  quantity, 

2.  Find  the  cost  of  |  yd.  of  silk  at  18  a  yd. 
Name  the  number,,  the  unit,,  and  the  quantity. 

3.  In  taking  |^  of  a  number  is  it  better  to  divide 
by  4  and  then  multiply  by  3,  or  to  multiply  first  by 
3  and  then  divide  by  4  ?     Test  in  these  questions  : 


f  of  20  =  ? 

1  of  15  =  ? 

fofT    =? 

1  of  13  =  ? 

f  of  32  =  ? 

f  of  48  =  ? 

Find  the  cost  of : 

4.    10  lb.  sugar  at  SJ^  a  lb. 

6  bu.  potatoes  at  If  a  bu 

,. 

5  baskets  melons  at  $  g^^  a  basket. 

5.    1  doz.  5-lb.  jars  butter  at  $  1  Jg-  each. 

fyd.  cloth  at  12  a  yd. 

2  doz.  yd.  towelling  at  $ 

i  a  yd. 

Find  the  weight  of  : 

6.    8  yd.  carpet  at  1\  lb.  a 

yd.;  lilb. 

1 A  lb.;  if  lb. 

iiV  lb.; 


81 


82  MENTAL  ARITHMETIC 

7.  8  double  rolls  of  wall  paper  at  1|  lb.  each ;  6 
double  rolls  ;  12  ;   7  ;  9  ;  10. 

8.  3  spoons  at  1^  oz.  each  ;  5  at  |  oz.  each  ;  |  doz. 
at  I  oz.  each ;  -|  doz.  at  1|  oz.  each. 

9.  Half  a  dozen  teaspoons,  each  4|^  in.  long  are 
placed  end  to  end ;  how  long  a  line  do  they  make  ? 
How  long  if  each  is  5|-  in.  long?     5|  in.? 

10.  How  many  ounces  of  pure  silver  in  a  dozen 
tablespoons  weighing  1|  oz.  each,  |^  fine  ? 

11.  Reduce  to  inches  :  fft.;  l^ft.;  2ift.;  3|  ft. 
f  ft.;  fft.;   If  ft. 

12.  Reduce  to  yards  :  2  rd.;  4  rd. ;  6  rd. ;  5  rd. 
9  rd. ;  10  rd. ;  12  rd. 

13.  Reduce  to  feet :    2  rd. ;  6  rd. ;  3  rd. ;  10  rd. 
12  rd. 

14.  Reduce  to  quarts  :  1|  gal. ;  |  gal. ;  1|  gal. 
2igaL;   3^5^  gal.;  U^\  g^h 

15.  Reduce  to  square  feet :    |  sq.  yd. ;    ^  sq.  yd. 
4  sq.  yd.;   21  sq.  yd. 

16.  Reduce  to  ounces  avoirdupois :    |^  lb. ;    J  lb. 
I  lb.;  211b.;  511b. 

17.  What  is  the  area  of  a  table  3  ft.  long  and 
21  ft.  wide  ? 

18.  Find  the  areas  of  the  floors  of  the  following 
rooms : 

Length  Width  Length  Width 

8  yd.         6|yd.  10  yd.         7f  yd. 

9  yd.         7|yd.  12  yd.         8f  yd. 
7  yd.         5Jyd.  9  yd.         6|  yd. 


LESSON  29  88 

19.  Find  the  areas  of  the  walls  of  the  following 
rooms  : 

Perimeter         Height  Perimeter  Height 

60  ft.       8  ft.  6  in.  80  ft.       10  ft.  4  in. 

80  ft.       9  ft.  9  in.  90  ft.         9  ft.  4  in. 

20.  The  base  of  a  triangle  is  12  ft.  and  the  alti- 
tude 6  ft.  6  in.     Find  its  area.      (See  p.  35,  ex.  2.) 

21.  Reduce  2|  and  4^  to  improper  fractions. 
iof4i  =  ?        8x|=?        2fx4i=? 

22.  How  do  you  find  the  product  of  two  numbers 
like  2f  and  41? 

23.  Find  the  value  of : 


11-  X  H 

7ix2^V 

3f  xlf 

fx    f 

lfx9i 

fx    f 

2|x4i 

2^x5i 

fx    1 

8|x4| 

24.    Find  the  value  of : 

Six  51 

^of    |ofl2 

l|x2i 

fof    |of24 

18J  X  3f 

fof    fof  36 

12    xlf 

41  of    -fof  15 

2fx6| 

f  of  4f  of  27 

131  X  4i 

1  of  6^  of  8 

25.    If  a  man  earns  $2| 

a  day,  how  much  will  he 

earn  in  3|  da.? 

26.    Find  the  volume  of  { 

I  rectangular  slab  of  gran- 

ite  41  ft.  long,  21  ft. 

wide, 

and  1^  ft.  thick. 

84  MENTAL  ARITHMETIC 

27.  What  is  the  weight  in  pounds  and  ounces  of  a 
square  piece  of  floor  oil  cloth  1|  yd.  long,  weighing 
3|  lb.  to  the  square  yard  ? 

28.  What  is  the  weight  of  a  mat  27  in.  wide, 
48  in.  long,  which  weighs  21  lb.  to  the  square  yard  ? 

Lesson  30 

1.  If  1  bu.  of  corn  is  worth  $|,  what  is  |  bu. 
worth  ? 

2.  What  will  51  T.  of  coal  cost  at  |5^  a  ton  ? 

3.  What  will  a  5-gal.  can  of  kerosene  cost  at 
121^  a  gal.? 

4.  How  many  square  miles  are  there  in  a  section 
of  country  4|-  mi.  long  and  2|  mi.  wide  ? 

5.  A  piece  of  park  is  in  the  form  of  a  triangle. 
If  its  base  is  7^  rd.  and  its  altitude  6|  rd.,  what  is 
its  area?     (See  Lesson  13,  example  2.) 

6.  A  lot  in  the  form  of  a  trapezoid  is  9^  rd.  wide 
in  front  and  6^  rd.  wide  in  the  back.  If  its  depth 
is  10|-  rd.,  what  is  its  area?  (See  Lesson  13,  ex- 
ample 15.) 

7.  How  far  will  a  person  travel  in  3^  hr.  at  5^  mi. 
an  hr.  ? 

8.  How  many  yards  in  1  rd.  ?  How  many  square 
yards  in  1  sq.  rd.  ? 

9.  At  $1  a  yd.,  what  part  of  a  dollar  will  |  yd. 
cost  ? 


LESSON  30  85 

10.  A  farmer  can  plough  a  Held  in  18  hr.  What 
part  of  it  can  he  plough  in  2  hr.  ?  In  3|  hr.  ?  In 
2  hr.  15  min.  ?     In  6  hr.  45  min.  ? 

11.  If  I  of  I  of  1^  of  a  board  20  ft.  long  is  cut  off, 
what  length  of  board  is  left  ? 

12.  How  much  greater  is  -fj  of  2|  rd.  than  1  yd.  ? 

13.  Texas  is  4J  times  as  large  as  Illinois,  and 
Illinois  is  2^  as  large  as  West  Virginia.  How 
many  times  as  large  as  West  Virginia  is  Texas? 
What  part  of  Texas  is  as  large  as  West  Virginia  ? 

14.  If  Michigan  is  |^  as  large  as  California  and 
Oklahoma  is  |  as  large  as  Michigan,  compare  the 
area  of  Oklahoma  with  that  of  California. 

15.  If  the  population  of  the  District  of  Columbia 
is  f  that  of  Rhode  Island,  how  many  people  live  in 
the  District  of  Columbia  for  every  99  who  live  in 
Rhode  Island  ? 

16.  If  the  population  of  Utah  is  J  that  of  Colorado, 
and  that  of  Colorado  ^  that  of  Michigan,  how  many 
people  live  in  Utah  for  every  100  who  live  in  Michi- 
gan ?  How  many  people  in  Michigan  for  every  100 
in  Utah  ? 

17.  The  Mackenzie  River  is  ^  as  long  as  the  Ama- 
zon, and  the  Rio  Grande  ^^  as  long  as  the  Mac- 
kenzie. How  long  is  the  Rio  Grande,  the  Amazon 
being  4000  mi.  in  length  ? 


86  MENTAL   ARITHMETIC 

18.  I  bought  a  house  and  lot  and  paid  |  of  the 
cost  price  in  cash?  If  this  was  $3600,  what  was 
the  cost  price  ? 

19.  A  man  sold  his  horse,  gaining  |  of  the  cost 
price.  If  his  gain  was  124,  what  did  the  horse  cost 
him  ? 

20.  A  man  owned  a  |  interest  in  a  factory  and 
sold  "I  of  his  share.  What  part  of  the  factory  did 
he  sell  and  what  part  did  he  keep  ? 

21.  A  man  invested  f  of  his  money  in  a  house. 
What  part  of  his  money  was  left  ?  If  he  invested 
|-  of  this  remainder  in  business,  what  part  of  his 
money  did  he  invest  in  business?  If  this  was 
$2400,  how  much  money  had  he  at  first  ? 

22.  A  man  spent  -^j  of  his  money  for  a  farm  and 
I  of  the  remainder  for  cattle.  If  he  paid  11800  for 
the  cattle,  how  much  money  had  he  at  first  ? 

23.  A  merchant  bought  goods  to  the  amount  of 
1360,  but  he  got  J  off  as  a  merchant,  and  then  2V  off 
the  remainder  by  paying  cash.  What  did  the  goods 
cost  him  ? 

24.  A  wholesale  merchant  sold  goods  amounting 
to  1600,  but  he  deducted  ^  of  the  amount  of  the  bill 
and  then  threw  off  J^  of  the  remainder  for  cash. 
Find  the  net  selling  price. 

25.  A  wholesale  merchant  threw  off  ^  of  the 
amount  of  a  bill  and  then  ^  of  the  remainder,  and 
the  bill  then  amounted  to  $300.  What  was  the 
original  bill? 


LESSON  31  87 

26.  A  piece  of  cloth  when  measured  with  a  yard- 
stick that  is  I  in.  too  long  seems  to  be  10|-  yd.  long. 
How  many  inches  longer  than  that  is  it?  How 
many  yd.,  ft.,  and  in.  long  is  it  ? 

Lesson  31 

1.  Into  how  many  parts,  each  2  ft.  long,  can  you 
divide  a  line  6  ft.  long?     6  f t.  -^  2  ft.  =  ? 

2.  Imagine  a  line  1  ft.  long  divided  into  parts 
each  J  ft.  long.     How  many  parts  ?     1  ft.  ^  |^  ft.  =  ? 

1  ft.   ^1  ft.   =  ? 
1  ft.   ^  1   ft.   =  ? 

IT.  "f-  ^  T.   =  ? 
1  bu.  -  J^  bu.  =  ? 

5.  Into  how  many  lots,  each  equal  to  |^  A.,  can 
you  divide  1  A.  ? 

6.  A  society  distributed  1  T.  of  coal  to  several 
families,  giving  to  each  ^  T.  How  many  families 
were  there? 

7.  Into  how  many  parts,  each  ^  ft.  long,  can  you 
divide  a  line  1  ft.  long?     4  ft.  long  ?     6  ft.? 

6  ft.    -r-  i  ft.    =  ?  4  ft.   ^  i  ft.   =  ? 


3.    1ft. 

-5-  i  ft.  =  ? 

1ft. 

^ift.=? 

4.    lib. 

-^llb.  =  ? 

1  A. 

^iA.  =  ? 

'S 


8  yd.  H-|yd.  =?  9  hr 

7  gal.  H- 1  gal.  =  ?  5  bu.  -^  f ^  ba.  =  ? 


■J 


88  MENTAL  ARITHMETIC 

8.  A  five-gallon  can  of  maple  syrup  is  emptied 
into  jars,  each  containing  ^  gal.  Find  the  number 
of  jars. 

Name  the  quantity,  the  unit,  and  the  number. 

9.  A  girl,  working  problems  in  arithmetic,  did  2 
in  every  ^  hr.     How  many  did  she  do  in  2  hr.  ? 

10.  Into  how  many  parts,  each  1  ft.  long,  can  you 
divide  a  line  6  ft.  long?  How  many  of  these  18 
parts  make  ^  ft.  ?  How  many  parts,  each  equal  to 
I  ft.,  can  you  count  in  these  18  parts. 

6  ft.  -^  I  ft.  =  ?  4  ft.  -^  I  ft.  =  ? 

11.  In  the  first  example  in  question  10  by  what 
did  you  multiply  6  to  get  18?  By  what  did  you 
divide  18  to  get  9?  Is  it  correct  to  divide  by  2 
first,  and  then  multiply  by  3?  Which  gives  the 
smaller  numbers? 

12.  How  do  you  divide  a  whole  number  by  a 
fraction  ? 

13.  Find  the  quotient  in  each  of  the  following : 


6  ft. 

-f  ft. 

$4- If 

8  ft. 

-^  f 

10  yd. 

^fyd. 

|8^fA 

20  yd. 

^i 

6  1b. 

-f-f  lb. 

110^  If 

141b. 

^i 

4T. 

^fT. 

fl2^$f 

18  T. 

^^ 

10  A. 

H-f  A. 

-114-131 

20  A. 

-^H 

LESSON  31  89 

14.  How  many  strips  of  Brussels  carpet  |  yd. 
wide  are  needed  to  carpet  a  room  6  yd.  wide?  If 
this  room  is  8  yd.  long,  how  many  yards  of  carpet 
will  be  needed  for  the  room,  provided  there  is  no 
waste  in  matching?  How  many  yards  will  be  needed 
if  there  is  a  waste  of  |  yd.  on  each  strip  in  matching  ? 

15.  How  many  strips  of  hemp  carpet  f  yd.  wide 
are  needed  to  carpet  a  room  8  yd.  wide?  If  the 
room  is  10  yd.  long,  how  many  yards  of  carpet  are 
needed  for  it? 

16.  What  part  of  a  yard  is  18  in.  ?  How  many 
widths  of  wall  paper  18  in.  wide  will  be  needed  to 
paper  a  room  whose  perimeter  is  20  yd.  ?  A  room 
whose  perimeter  is  90  ft.  ? 

17.  What  part  of  a  yard  is  22  in.  ?  How  many 
widths  of  wall  paper  22  in.  wide  will  be  needed  to 
paper  a  room  6  yd.  long  and  5  yd.  wide? 

18.  6-^1  =  ?  8^t  =  ?  fx|  =  ? 

How  do  you  divide  one  fraction  by  another  ? 

19.  f 

1    . 

2 

3.. 
5 


What  is  the  ratio  of  J  to  f  ?     Of  IJ  to  2|?     Of  21 
to  U? 


I=? 

I^^=? 

i^f  =? 

I=? 

l-t=? 

2J^if  =  ? 

f=? 

f^T\  =  ? 

3J-^lf  =  ? 

J=? 

io^^=^ 

5|^7i=? 

90  MENTAL  ARITHMETIC 


20.    31^51=? 

H^H  =? 

6|^2|  =  ? 

4|^3i  =  ? 

2f^6J   =? 

7|^2i=? 

ii-if  =  ? 

9i^lxV=? 

1|-2|  =  ? 

What  is  the  ratio  of  4^  to  51?  Of  6  to  16 J?  If 
to4f? 

21.  How  many  rods  in  22  yd.  ?  In  33  yd.?  Re- 
duce to  the  fraction  of  a  rod :  2|  yd.,  1|  yd.,  3^  yd., 
7iyd.,l^yd. 

22.  Reduce  to  the  fraction  of  a  foot :  2|  in.,  7^ 
in.,  3f  in.,  6f  in.,  2|  in. 

23.  Reduce  to  the  fraction  of  a  yard  :  1  ft.  6  in. ; 
2  ft.  3  in.  ;  2  ft.  6  in. 

24.  Reduce  to  fractions  of  100  :  111  9_i_,  142^ 
22|,  44-1,  55-|,  33i,  66f,  61,  6f. 

25.  Barbed  wire  weighs  l^^g  lb.  to  the  rod.  How 
long  is  a  roil  of  wire  weighing  S^  lb.? 

26.  A  double  roll  of  wall  paper  weighs  If  lb. 
How  many  double  rolls  weigh  11^  lb.? 

27.  If  carpet  weighs  20  oz.  to  the  yard,  how  many 
yards  of  carpet  will  weigh  S2^  lb.? 

'28.    How  many  silver  dessert  spoons  weighing  15 
oz.  per  doz.  will  weigh  S^  lb.? 

29.  How  many  silver  spoons,  each  5|  in.  long, 
placed  end  to  end  wilt  make  a  line  5  ft.  9  in.  long  ? 


LESSON  32  91 

Lesson  32 

In  examples  1  and  2  the  statements  of  balance  are : 

(1)  The  rate  =  27  mi.  -  2^ 
Or                 2J  times  the  rate  =  27  mi. 

(2)  The  actual  distance  =  12  yd.  -  J^  in. 

Give  the  statements  of  balance  for  the  following  examples 
in  this  lesson. 

1.  A  bicyclist  rode  27  mi.  in  2  hr.  15  min.  What 
was  his  rate  in  miles  per  hour  ? 

2.  If  a  yard  measure  is  ^  in.  too  short,  what  is  the 
actual  distance  between  two  points  which  is  found  by 
this  measure  to  be  12  yd.  ? 

3.  After  pumping  out  360  gal.  of  water  from  a 
cistern,  |  of  the  water  still  remained.  How  many 
gallons  did  the  cistern  contain  at  first  ? 

4.  After  selling  400  bu.  of  wheat,  a  farmer  had 
f  of  his  crop  left.  How  many  bushels  remained 
unsold  ? 

5.  What  part  of  a  mile  is  440  yd.?  30  sec.  is 
what  part  of  1  min.  ?  Of  1  hr.  ?  A  train  travels 
440  yd.  in  30  sec.  This  is  at  the  rate  of  how  many 
miles  an  hour  ? 

6.  Railway  fencing  costs  $5.50  a  rod.  What  is 
the  cost  of  1  yd.  of  fencing  ?     Of  2  mi.  ? 

7.  Find  the  number  which  is  35  more  than  the 
sum  of  J  and  ^  of  it.     Prove  your  answer  correct. 


92  MENTAL  ARITHMETIC 

8.  If  to  a  certain  number  its  ^  and  ^  be  added, 
the  sum  will  be  66,  Find  the  number.  Prove  your 
answer  correct. 

9.  A  man  paid  out  f  of  the  money  in  his  pocket 
to  settle  a  grocery  bill,  |  of  the  remainder  to  settle 
a  meat  bill.  If  he  had  $9  left,  how  much  money 
had  he  at  first  ?     Prove  your  answer  correct. 

10.  A  lady  spent  ^  of  the  money  in  her  purse  for 
a  jacket,  and  ^  of  the  remainder  for  a  hat.  If  the 
jacket  cost  $12  more  than  the  hat,  how  much  money 
did  the  purse  contain  at  first  ?  Prove  your  answer 
correct. 

11.  One-fifth  of  the  number  of  pupils  in  a  high 
school  are  in  the  highest  class,  and  |  of  the  remain- 
der in  the  lowest  class.  If  there  are  5  more  pupils 
in  the  middle  than  in  the  highest  class,  how  many 
pupils  are  in  the  school  ?    Prove  your  answer  correct. 

12.  A  man  bought  a  house  and  lot;  ^  of  their 
value  was  equal  to  4  times  the  cost  of  repairs.  The 
total  cost  was  f  2600.  Find  the  cost  of  the  house  and 
lot. 

13.  If  gun  metal  is  composed  of  11  parts  of  copper 
to  2  parts  of  tin,  how  many  pounds  of  each  in  65  lb. 
of  gun  metal  ? 

14.  A  room  is  ^  again  as  long  as  it  is  wide.  If 
its  perimeter  is  84  ft.,  what  are  its  dimensions  ? 


LESSON  32  93 

15.  The  Hudson  River  is  ^  as  long  as  the  Red 
River,  and  the  Potomac  is  |  as  long  as  the  Hudson. 
The  Potomac  River  is  350  mi.  in  length.  How  long 
is  the  Red  River  ? 

16.  The  Connecticut  River,  which  is  450  mi.  long, 
is  \  shorter  than  the  Cumberland,  and  the  Cumber- 
land ^  as  long  as  the  Nelson.  How  long  is  the  Nelson 
River  ? 

17.  A  man  invested  |  of  his  money  in  business, 
but  it  having  proved  a  failure  he  sold  it  for  |  of 
what  he  invested  in  it.  If  his  loss  was  $  2700,  how 
much  money  had  he  at  first  ? 

18.  A  man  left  ^  of  his  property  to  his  wife,  ^  of 
the  remainder  to  his  daughter,  and  the  rest  to  his 
son.  What  was  the  share  of  each,  and  what  was  the 
property  worth  if  the  son  received  $  6000  more  than 
the  daughter  ? 

19.  A  man  earns  three  times  as  much  as  his  oldest 
son,  and  the  oldest  son  earns  twice  as  much  as  his 
brother.  What  part  of  the  family  income  does  each 
earn?  If  this  is  $45  a  week,  what  are  the  weekly 
earnings  of  each  ? 

20.  A  speculator  sold  wheat  at  the  rate  of  30  bu. 
for  $25,  and  by  doing  so  lost  ^  of  the  cost  price. 
What  did  the  wheat  cost  him  per  bu.? 

21.  What  is  the  ratio  of  the  cost  of  1^  T.  of  coal 
to  that  of  31  T.  ?  If  31  T.  of  coal  cost  $  21,  what 
will  If  T.  cost  ? 


94  MENTAL  ARITHMETIC 

22.  I  sold  a  horse  for  1 60  and  thereby  lost  ^  of 
the  cost  price.  What  should  I  have  sold  it  for  to 
gain  ^  of  the  cost  ? 

23.  A  boy  bought  3  doz.  eggs  for  his  mother  at 
12^  a  doz.,  and  with  the  remainder,  which  was  ^  of 
the  money  she  gave  him,  he  bought  oranges  at  4)^ 
each.     How  many  oranges  did  he  buy  ? 

24.  Three  farms  join  each  other.  The  first  con- 
tains |-  as  much  as  the  second,  and  the  second  -|  as 
much  as  the  third.  The  third  farm  contains  120  A. 
How  large  is  the  first  farm  ? 

25.  I  traded  a  f  interest  in  a  grocery  store  for  a 
•|  interest  in  a  general  store.  If  the  grocery  store 
was  valued  at  $9000,  what  was  the  value  of  the 
general  store? 


DECIMALS 

Lesson  33 

1.  In  the  expression  $6.37  the  6  is  6  what?  The 
3  is  3  what?  What  part  of  a  dollar?  The  7  is  7 
what  ?  What  part  of  a  dollar  ?  The  37  Is  37  what  ? 
What  part  of  a  dollar  ? 

2.  In  the  expression  $2.19  what  part  of  a  dollar 
is  the  1  ?     The  9  ?     The  19  ? 

3.  Read  as  dollars  and  as  hundredths  of  a  dollar: 
$4.29,  $6.37,  $2.15,  $8.05,  $8.99,  $.76,  $.26,  $.06, 
$.09. 

4.  Read  the  following  expressions  : 

3.45  T.,  2.96  mi.,  7.15  A.,  6.43  yr.,  .37  da.,  .89  hr., 
.07  hr.,  .08  gal.,  3.06  gal.,  5.01  bu.,  3.18,  .64,  3.50, 
.04,  .02. 

Give  the  place  value  of  each  figure  in  the  above 
example. 

5.  Read  the  following  expressions  : 

3.146  lb.,  4.293  T.,  8.141  A.,  5.16  A.,  8.028  T., 
3.006  da.,  .456  da.,  .75  hr.,  .005  bu.,  .003  gal., 
4.568  mi.,  3.333,  4.593,  .063,  .008,  .06,  .006. 

Give  the  place  value  of  each  figure  in  the  above 
example. 


96  MENTAL  ARITHMETIC 

6.  Read  the  following  expressions  : 

.292,  6.04,  .6,  .64,  .645,  .6457,  .64572,  .6,  .06, 
.006,  .0006,  .0054,  94.3006,  .0016,  7.24732,  3.0306, 
12.47025,  .42819,  .00328,  .8216,  3.1416,  2150.42, 
39.371. 

7.  Reduce  to  common  fractions  in  their  lowest 
terms :  .5,  .6,  .20,  .25,  .75,  .80,  .125,  .375,  .625,  .875. 

8.  A  farmer  sold  .8  of  his  flock  of  325  sheep. 
How  many  did  he  sell  ?  Give  the  statements  of 
balance  for  this  and  the  following  examples. 

9.  A  farmer  gave  .375  of  his  farm  of  200  A.  to 
his  son.     How  many  acres  did  he  keep? 

10.  A  merchant  sold  cloth  at  an  advance  of  .125 
of  the  cost  and  gained  21  j^  a  yd.  What  did  a  cus- 
tomer pay  for  10  yd.? 

11.  Reduce  to  common  fractions  in  their  lowest 
terms:  .331    .66|,  .16f,  .Mf  .121   .371  .621  .871 

111       992       444        555.       771       GQ8        a2        ai        081 

Commit  these  results  to  memory. 

12.  A  man  withdrew  .441  of  the  money  he  had  in 
a  bank  to  invest  it  in  business.  There  still  remained 
in  the  bank  $8500.     How  much  did  he  withdraw? 

13.  A  merchant  sent  .06|  of  a  piece  of  cloth,  which 
originally  contained  45  yd.  and  sold  at  50^  a  yd.,  to 
the  remnant  counter,  reducing  the  price  .5.  What 
was  the  piece  marked  to  sell  at? 


LESSON  33  97 

14.  A  merchant  bought  goods  from  a  wholesale 
house  for  which  he  received  a  bill  for  $480  less  .33 J 
and  .06^  for  cash.     Find  the  cash  amount  of  the  bill. 

15.  ,66^  of  a  piece  of  ribbon  was  sold  to  one  cus- 
tomer and  .16|  of  it  to  another.  The  remainder 
sold  for  36^  at  12  j^  a  yard.  Find  the  number  of 
yards  in  the  piece  at  first. 

16.  Reduce  to  hundredths  :  |^,  ^,  -|,  J,  f ,  |,  f ,  ^,  |-, 

A     1      1      3      5      7     1     2      4      5     1     i     JL     JL     _1_     JL     JL 
e**    y     8'    8^    8'    ¥'     9'    9'    9'    9'     9'     9'    10'    11'    12'    15'     16* 

Commit  these  results  to  memory. 

17.  What  part  of  1  yd.  is  1  ft.?  How  many  hun- 
dredths ?  What  part  of  1  ft.  is  1  in.  ?  How  many 
hundredths  ? 

18.  Express  your  answers  to  the  following  ques- 
tions in  tenths  or  hundredths.  What  part  of  1  gal. 
is  1  qt.?  Of  1  bu.  is  2  pk.?  Of  1  pk.  is  1  qt.?  Of 
1  lb.  avoir,  is  1  oz.?  Of  1  lb.  troy  is  1  oz.?  Of 
.25  of  1  yr.  is  1  mo.?  Of  1  da.  is  4  hr.?  Of  .16|  of 
1  quire  is  .33^  of  a  dozen  sheets  of  paper?  Of  1  rd. 
is  1  yd.?  Of  1  sq.  yd.  is  1  sq.  ft.?  Of  the  side  of 
a  square  containing  25  sq.  in.  is  the  side  of  a  square 
containing  16  sq.  in.? 

19.  Carpet  bought  at  56^  a  yd.  is  sold  for  70^ 
a  yd.  What  is  the  gain  as  a  fraction  of  the  cost? 
How  many  hundredths  ? 

20.  A  grocer  bought  tea  at  60^  a  lb.,  and  sold  it 
for  72^  a  lb.  How  many  hundredths  of  the  cost  did 
he  gain  ? 


98  MENTAL  ARITHMETIC 

21.  Bought  pencils  at  30  j^  a  dozen  and  sold  them 
at  4^  each.    What  is  the  gain  on  each  dollar  invested? 

22.  A  laboring  man  earns  1 270  in  240  da.  at  11.50 
a  day.  How  many  days  does  he  work  out  of  every 
100  da.? 

23.  A  person  invested  .33|^  of  his  money  in  real 
estate  and  .44^  of  it  in  business.  He  had  $1200  left. 
What  was  the  amount  of  each  investment? 

24.  A  grocer  sold  .16|  of  his  stock  of  eggs  for  24^ 
a  doz.,  .5  of  it  for  22^  a  doz.,  and  the  remainder  ISj^ 
a  doz.     Find  the  average  price  per  doz. 

25.  The  stock  in  a  drygoods  store  was  damaged 
.33^  of  its  value  by  fire.  If  the  stock  was  worth 
$  9000,  what  was  the  loss  of  one  of  the  partners  who 
had  a  |  interest  in  the  business  ? 


COMPOUND   QUANTITIES 

Lesson  34 

In  the  following  examples  give  the  statements  of 
balance  : 

1.  Find  the  quantity  measured  by  the  number  6 
and  the  unit  1  gal.  2  qt. 

2.  Mark  off  the  space  occupied  by  a  cord.  How 
many  cords  in  a  pile  of  cordwood  24  ft.  long  and 
4  ft.  high  ?  24  ft.  long  and  8  ft.  high?  What  is  the 
cost  of  the  latter  pile  at  $  4  a  cord  ? 

3.  How  many  acres  in  1  sq.  mi.  ?  What  part  of 
a  square  mile  is  a  farm  of  160  A.  ?  If  this  farm  is 
in  the  form  of  a  square,  what  is  the  length  of  its 
side  ? 

4.  How  long  will  it  take  to  walk  around  the 
farm  in  example  3  at  the  rate  of  3  mi.  an  hr.  ? 

5.  How  might  such  expressions  as  the  following 
arise  by  acts  of  measurement  :  2  rd.  3  yd. ;  3  bu. 
1  pk. ;  4  gal.  2  qt.  1  pt.  ;  5  lb.  6  oz. ;  2  T.  400  lb.  ; 
3  yr.  2  mo. ;  4  hr.  26  min.  ,  3  doz.  6  ? 

6.  What  part  of  1  yd.  is  2  ft.  ?  6  in.  ?  2  ft.  6  in.  ? 
Find  the  weight  of  an  iron  bar  5  yd.  2  ft.  6  in.  long 
if  1  yd.  weighs  15  lb. 

99 


loo  MENTAL  ARITHMETIC 

7.  How  many  pecks  in  1  bu.  ?  A  farmer  sowed 
5  pk.  of  oats  on  a  field  and  raised  16  bu.  1  pk. 
What  is  the  yield  per  peck  of  seed  ?  per  bushel  ? 

8.  A  train  travels  at  the  rate  of  ^  mi.  in  30  sec. 
What  is  its  rate  per  hour  ? 

9.  What  is  the  value  of  X 1  in  dollars  and  cents  ? 
A  gentleman  in  New  York  places  c£100  to  his  credit 
in  London.  What  will  it  cost  him,  exchange  being 
quotedat  $4.8T? 

10.  A  cubic  foot  of  water  weighs  1000  oz.  How 
many  pounds  ?  20  cu.  ft.  of  water  weigh  what  part 
of  a  ton  ? 

11.  Apples  are  bought  at  1 2  a  bbl.  of  2|  bu.,  and 
sold  at  30  j^  a  pk.  What  is  the  gain  per  bbl.  ?  What 
would  the  gain  be  if  1  pk.  in  each  barrel  were  spoiled  ? 

12.  How  many  square  rods  in  1  A.  ?  What  is  the 
value  of  a  triangular  piece  of  land  whose  base  is 
32  rd.  and  altitude  15  rd.  at  $60  an  acre  ?  (p.  35.) 

13.  How  much  ink  will  it  take  to  fill  the  ink-wells 
in  a  room  containing  48  single  desks,  if  each  well 
holds  I  gill  ? 

14.  How  many  pounds  does  1  bu.  of  wheat  weigh  ? 
What  is  the  cost  of  2  bu.  15  lb.  of  wheat  at  84^ 
a  bu.  ? 

15.  How  many  square  chains  in  1  A.  ?  What  is 
the  selling  price  of  a  piece  of  land  20  ch.  long  and 
16  ch.  wide  at  1 90  an  acre  ? 


LESSON  34  TOi 

16.  A  fruit  dealer  bought  a  bushel  of  apples  for 
$1.50,  and  sold  them  at  the  rate  of  3  for  5^.  How 
much  did  he  gain  if  there  were  240  apples  in  the 
bushel  ? 

17.  How  many  loads  of  earth  were  removed  to 
make  a  cellar  15  ft.  long,  12  ft.  wide,  and  6  ft.  deep  ? 

18.  How  many  lead  pencils  in  a  dozen  boxes  if 
each  contains  ^  gross  ? 

19.  When  apples  are  selling  at  40  ^  a  peck,  what 
profit  is  made  by  the  seller  who  buys  them  at  $  1.20 
a  bushel  ? 

20.  What  will  the  lumber  cost  for  a  tight  board 
fence  250  ft.  long  and  4  ft.  high  at  12  per  C  ? 

21.  A  square  rug  2|  yd.  long  costs  $  5  ;  what  is 
the  price  per  square  yard? 

22.  A  bottle  of  sachet  powder  weighs  6  oz.  What 
is  the  least  number  of  bottles  that  weigh  an  exact 
number  of  pounds  ? 

23.  How  many  pounds  of  silver  will  make  240 
teaspoons,  |^  fine,  if  a  dozen  spoons  weigh  6  oz.  ? 

24.  Each  of  the  two  sides  of  a  roof  is  50  ft.  long 
and  24  ft.  wide.  What  is  the  cost  of  iron  roofing  to 
cover  this  roof  at  12.50  per  square  of  100  sq.  ft.  ? 

25.  How  many  4-in.  cubes  can  be  cut  from  a 
block  of  wood  1  ft.  8  in.  long,  1  ft.  wide,  and  8  in. 
thick,  no  allowance  being  made  for  waste  ? 

26.  Find  the  weight  in  pounds  of  12  doz.  silver 
tablespoons  weighing  20  oz.  per  doz.  Of  12  yd.  of 
carpet  weighing  20  oz.  each. 


l{)2  MENTAL  ARITHMETIC 

Lesson  35 

1.  Find  the  value  of  the  quantity  measured  by 
the  number  8  according  as  the  unit  is  2  ft.  6  in.  or 
2  bu.  6  pk. 

2.  What  are  the  dimensions  of  a  cord  ?  Find 
the  value  of  a  pile  of  cordwood  32  ft.  long  and  6  ft. 
high  at  $3.50  a  cord. 

3.  What  part  of  a  rod  is  1  ft.  ?  5^  ft.  ?  22  ft.  ? 
The  driving  wheel  of  a  locomotive,  which  is  22  ft. 
in  circumference,  turns  90  times  in  1  min.  Through 
how  many  rods  does  it  draw  the  train  in  1  min.  ? 

4.  What  part  of  a  mile  is  40  rd.  ?  120  rd.  ?  160  rd.  ? 
280  rd.  ?  A  train  runs  120  rd.  in  1  min.  How 
many  miles  is  that  an  hour  ? 

5.  How  many  rods  in  1  mi.?  How  many  yards? 
feet  ?  A  horse  trotted  1  mi.  in  2  min.  b6  sec.  Tak- 
ing his  stride  at  5  yd.,  how  many  times  a  second  did 
his  feet  touch  the  ground  ? 

6.  What  part  of  an  hour  is  18  min.?  36  min.? 
45  min.?  48  min.?  54  min.?  A  horse  travels  24  mi. 
in  2  hr.  24  min.     What  is  the  rate  per  hour  ? 

7.  Reduce  f  of  1 48  to  a  decimal  of  $  80. 

8.  How  many  shillings  in  £1?  £15?  £1  16s.? 
£S  4s.?  £2  lis,?  How  many  ounces  of  gold  are 
worth  £21  if  .6  oz.  is  worth  £  2  14s.? 


LESSON  35  103 

9.  How  do  you  find  the  perimeter  of  an  oblong 
field  ?  How  many  rods  of  wire  fencing  will  it  take 
to  enclose  a  field  40  rd.  long  and  25  rd.  wide,  there 
being  5  rows  of  wire  ? 

10.  What  will  the  wire  cost  at  $  2  per  spool  of 
100  lb.,  if  wire  fencing  weighs  1  lb.  to  the  rod  ? 

11.  What  part  of  a  yard  is  18  in.?  How  many 
strips  of  wall  paper  18  in.  wide  will  be  required  to 
paper  a  room  20  ft.  long  and  16  ft.  wide,  no  allow- 
ance being  made  for  waste  in  matching  ? 

12.  If  the  room  in  the  previous  example  is  12  ft. 
high,  how  many  double  rolls  of  paper  will  be  needed 
for  the  room,  and  what  will  be  its  weight?  (A 
double  roll  weighs  1|  lb.  and  is  16  yd.  long.) 

13.  State  clearly  and  briefly  how  you  solved  each 
of  the  foregoing  examples  in  this  lesson.  State  how 
you  will  solve  each  of  the  following  examples  in  this 
lesson. 

14.  What  are  the  factors  of  231  ?  I  buy  molasses 
put  up  in  a  tin  dish  10^'  by  1'^  by  21  in.  What  part 
of  a  gallon  do  I  get? 

15.  Give  the  table  of  Troy  weight.  Find  the 
weight  in  pounds  of  one  dozen  silver  teaspoons 
weighing  |  oz.  each.  Find  their  cost  at  50^  each 
if  3  initials  are  engraved  on  each  spoon  at  a  charge 
of  l|y  a  letter. 


104  MENTAL   ARITHMETIC 

16.  A  man  paid  96^  for  1  lb.  of  tea,  but  on  weigh- 
ing it  found  that  it  was  ^  oz.  too  light.  What  should 
he  have  paid  for  it? 

17.  What  is  1  ft.  board  measure?  A  lot  250  ft. 
long  and  150  ft.  wide  is  to  be  surrounded  by  a  close 
board  fence  5  ft.  high.  What  will  the  lumber  cost 
at  113  per  M? 

18.  A  watch  gains  3  sec.  in  12  hr.  In  how  many 
days  will  it  gain  5  min.? 

19.  How  many  acres  in  1  sq.  mi.?  ^  sq.  mi.? 
1^  sq.  mi.?  How  many  square  miles  in  a  strip  of  land 
20  mi.  long  and  ^  mi.  wide?  20  mi.  long  and  -^-^  mi. 
wide?     How  many  acres  in  the  last  strip? 

20.  Whatpartof  Imi.  is880yd.?  88  yd.?  22  yd.? 
176  yd.? 

A  railway  company  pays  $20  an  acre  for  a  strip 
of  land  44  yd.  wide  and  20  mi.  long.  Find  the  entire 
cost. 

21.  Name  the  months  that  have  30  days ;    31  days. 
A  note  drawn  on  Aug.  27  is  due  Oct.  15.     How 

many  days  has  it  to  run? 

22.  A  note  due  Dec.  8,  1898,  was  discounted  Oct. 
16.  Find  the  number  of  days  between  the  day  of 
discount  and  the  day  on  which  the  note  is  due. 

23.  Through  how  many  degrees  does  the  sun  pass 
in  1  hr.?  How  do  you  change  longitude  expressed 
in  time  to  longitude  expressed  in  degrees? 


LESSON  36  105 

24.  A  man  travels  until  his  watch  is  2  hr.  4  min. 
fast.  Does  he  travel  east  or  west,  and  through  how 
many  degrees  ? 

25.  How  many  square  chains  in  1  A.?  How  many 
links  in  a  chain? 

What  is  the  value  of  land  24  ch.  50  1.  long  and 
20  ch.  wide  at  $60  an  acre? 

26.  Two  clocks  are  together  at  Monday  noon ;  one 
loses  10  sec.  and  the  other  gains  20  sec.  a  day.  In 
how  many  days  will  one  be  10  min.  ahead  of  the 
other?     On  what  day  of  the  week? 

Lesson  36 

1.  How  many  pounds  of  flour  are  there  in  a  sack 
containing  ^  bbl.?     -^  bbl.? 

2.  How  many  grains  in  1  lb.  Avoir.?  In  1  lb. 
Apothecaries'  ?  Do  druggists  use  Avoirdupois  or 
Apothecaries'  weight  in  buying  drugs?  In  selling 
drugs  ? 

3.  A  druggist  buys  1  lb.  of  quinine  for  $4  and 
uses  it  to  make  2  gr.  quinine  pills  which  he  sells  at 
30^  a  hundred.     How  much  does  he  gain? 

4.  Give  the  table  of  Apothecaries'  liquid  meas- 
ure. How  many  minims,  or  drops,  will  fill  a  tea- 
spoon? A  physician  prescribes  a  4  oz.  bottle  of 
medicine  to  be  taken  in  teaspoonful  doses  three  times 
a  day.  How  many  days  will  it  last  if  a  teaspoon 
holds  60  drops? 


106  MENTAL   ARITHMETIC 

5.  What  does  1  cu.  ft.  of  water  weigh?  How 
many  pounds  do  8  cu.  ft.  of  water  weigh?  What 
part  of  a  ton? 

A  cylindrical  column  of  water  whose  base  is  4  sq. 
ft.  and  height  8  ft.  weighs  how  many  tons  ? 

6.  How  many  board  feet  in  8  scantlings  12  ft.  by 
4  in.  by  3  in.? 

7.  Find  the  cost  of  5  bu.  3  pk.  2  qt.  of  nuts  at 
$1.28  a  bu.? 

8.  How  many  hours  of  daylight  are  there  on 
Aug.  18,  the  sun  rising  at  5.10  a.m.? 

9.  How  many  sheets  in  a  quire  ?  Quires  in  a 
ream?  A  stationer  pays  $1.56  a  ream  for  letter 
paper  and  sells  it  at  15  j^  a  quire.  How  much  does 
he  gain  on  1  ream  ?     This  is  what  part  of  the  cost  ? 

10.  Show  by  measuring  that  the  circumference  of 
a  circle  (plate,  etc.)  is  about  3^  times  the  diameter. 

.  How  many  feet  will  a  bicycle  having  a  28-in.  wheel 
go  in  making  12  revolutions? 

11.  How  often  does  the  unit  88  ft.  measure  the 
quantity  1  mi.? 

How  many  revolutions  will  a  28-in.  bicycle  wheel 
make  in  going  1  mi.?  Also  the  pedal,  the  gear 
being  3  ? 

12.  What  is  the  ratio  of  the  weight  of  1  lb.  of 
brass  to  that  of  1  lb.  of  gold  ?  How  many  pounds 
of  gold  weigh  as  much  as  72  lb.  of  brass? 


LESSON  36  107 

13.  If  14  yd.  of  cloth  30  in.  wide  will  make  a 
dress,  how  many  yards  of  cloth  28  in.  wide  will  be 
needed  to  make  another  dress  of  the  same  size  ? 

14.  An  ingrain  rug  30  in.  by  54  in.  cost  50  ^. 
What  was  the  price  per  square  yard  ? 

15.  A  man  rowed  4  miles  down  stream  in  40  min., 
and  back  again  up  stream  in  1  hr.  Find  his  rate  of 
rowing  per  hour  down  stream,  up  stream,  in  still 
water,  and  also  the  rate  of  the  current. 

16.  If  paint  brushes  that  cost  4  ^  each  cost  f  as 
much  when  bought  by  the  dozen,  what  is  the  cost  of 
6  doz.? 

17.  The  border  of  an  oblong  Moquette  carpet  is 
22^  in.  wide.  How  much  longer  is  the  perimeter  of 
the  border  on  the  outside  than  on  the  inside  ? 

18.  In  the  centre  of  a  square  room  12  yd.  on  a 
side  is  a  square  rug  covering  ^  of  the  floor.  Find 
the  length  of  its  side. 

19.  If  6  double  rolls  of  wall  paper  18  in.  wide 
are  needed  to  paper  a  room,  how  many  double  rolls 
of  22-in.  paper  would  paper  a  room  of  the  same  wall 
area  ? 

20.  What  is  the  ratio  of  the  side  of  a  6-in.  square 
to  that  of  a  9-in.  square  ?  Of  the  perimeter  ?  Of  the 
area  ? 

21.  What  is  the  ratio  of  the  side  of  a  square  rug 
1^  yd.  long  to  that  of  a  square  rug  \\  yd.  long  ?  Of 
•the  perimeter  ?     Of  the  area  ? 


108  MENTAL  ARITHMETIC 

22.  If  the  larger  rug  in  example  21  costs  $1.44, 
what  will  the  smaller  rug  cost  at  the  same  rate  per 
square  yard  ? 

23.  A  stationer  buys  lead  pencils  at  90  ^  per  half 
gross,  and  sells  them  at  the  rate  of  2  for  5  ^.  Find 
his  gain  on  1  gross. 

24.  How  many  quarts  in  1  bu.?  If  a  horse  eats 
1  pk.  3  qt.  of  oats  in  1  da.,  in  how  many  days  will 
it  eat  6  bu.? 

25.  A  township  6  mi.  square  is  divided  into  how 
many  sections  ?  What  is  the  area  of  one  section  in 
square  miles  ?  In  acres  ?  A  farmer  who  owns  ^^  of  a 
section  of  land  has  how  many  acres  in  his  farm  ?  If 
his  farm  is  in  the  form  of  a  square,  what  is  the  dis- 
tance around  it  ? 

26.  25  bu.  of  grain  occupy  32  cu.  ft.  A  bin  8  ft. 
by  4  ft.  by  4  ft.  contains  3200  lb.  of  grain.  Find 
the  weight  of  1  bu.  of  this  grain.  What  kind  of 
grain  is  it  ? 


PERCENTAGE 

Lesson  37 

1.  A  quantity  divided  into  thirds  is  measured  by 
3  units.  By  how  many  units  is  a  quantity  measured 
that  is  divided  into  fourths  ?  Into  fifths  ?  Fortieths  ? 
Sixtieths  ?  Hundredths  ? 

2.  A  quantity  considered  in  percentage  is  meas- 
ured by  100  units,  ^  of  the  quantity  by  ^  of  100  or 
25  units.  Thus  ^  of  a  quantity  is  25  hundredths  or 
25  per  cent  (25%)  of  it. 

3.  What  per  cent  of  a  quantity  is  |^  of  it  ?  ^?  -^^7 

1    ?    2?    3? 

^0  •     5  •    ^• 

4.  What  is  meant  by  saying  that  i  =  50  %  ? 
1=25%?  1=20%? 

5.  Express  the  following  fractions  in  per  cent : 

r^\    1134.    12345.     1_3__7        9._1_3 
K^J    ^^  2'  ¥'  ¥  '    5^'  "5'   5'  "g^'  "5  '  TO'  TOV  10'   10  '    2  0'  2  0"' 

_9_    11  .      1      _3      13     24 

20'  20  '    2^'  2^'   2  5'  2  5* 

rh^    123.15.1     3     67.    124:^     18.    JL. 
vV     3'   3'   3  '    6'   6  '    ^'  ^'   8'   8  '     9'   9'   9'   9'    9'   9  '     11  ' 
1.1.1 
12  '    T 5^  '     16* 

6.  What  is  the  ratio  of  1  qt.  to  1  gal.?  What 
per  cent  is  1  qt.  of  1  gal.?  What  is  the  ratio  of  1 
ft.  to  1  yd.?     What  per  cent  is  1  ft.  of  1  yd.? 

109 


110  MENTAL  ARITHMETIC 

7.  What  per  cent  is  1  pt.  of  1  qt.?  1  dime  of  $1? 
1  da.  of  1  wk.?  1  in.  of  1  ft.?  1  ft.  of  1  fathom? 
1  sq.  ft.  of  1  sq.  yd.?  1  sq.  ch.  of  1  A.?  1  oz.  of  1  lb. 
Avoir.?  1  oz.  of  1  lb.  Troy?  1  qt.  of  1  pk.?  1  sh. 
of  ^1? 

8.  What  per  cent  is  2  in.  of  1  ft.?  1  yd.  of  1  rd.? 
A  3-ft.  square  of  1  sq.  yd.?  A  2-ft.  square  of  a  3-ft. 
square?  8  oz.  of  2J  lb.  Avoir.?  30  min.  of  2  hr.? 
4  units  of  2  doz.?  1  dm.  of  1  m.?  1  qr.  of  1  rm.? 
77  cu.  in.  of  1  gal.?  1  rd.  of  1  ch.? 

9.  What  is  the  ratio  of  $3  to  $4?  What  per 
cent  is  $3  of  |4?  What  is  the  ratio  of  1 200  to 
1 600  ?     What  per  cent  is  $  200  of  1 600  ? 

10.  What  per  cent  of  $2  is  50^  ?  Of  $64  is  $24? 
Of  840  is  14^?  Of  48^  is  6^?  Of  $56  is  $49?  Of 
$225  is  $45?  Of  $2500  is  1 400?  Of  $5.28  is  $1.32? 
Of  $5  is  $2.45?   Of  $3  is  $1.25?   Of  $1  is  59^? 

11.  Memorize  the  following  table  : 

20%=i      331%  =  ^      50    %  =  l      22|%=| 
40%=f      66|%  =  f      62J%=f      16f%  =  J 
60%  =  f      12J%  =  i      75   %=f       Si%  =  j\ 
80%=|      25    fo  =  l      871%=!        6|%  =  ^V 
100%  =1       371%  =  !      111%  =i        6i%=J^ 
What  is  meant  by  saying  that  6 J  %  =  y^  ? 
In  the  following  examples  give  the  statements  of 
balance  : 

12.  A  grocer  bought  tea  at  75^  a  lb.,  and  sold  it 
at  a  gain  of  15  ^  a  lb.  What  is  the  ratio  of  the  gain 
to  the  cost  ?    What  per  cent  of  the  cost  did  he  gain  ? 


LESSON  37  111 

13.  A  merchant  bought  flour  at  $6  a  bbl.,  and 
sold  it  for  $  7.50  a  bbl.     Find  his  gain  per  cent. 

14.  Bought  cloth  at  80  ^  a  yd.,  and  sold  it  for  70  ^ 
a  yd.     Find  the  loss  per  cent. 

15.  At  the  end  of  the  season  the  following  articles 
were  advertised  in  the  Chicago  papers  as  marked 
down  in  price.  Find  in  each  case  the  rate  per  cent 
of  reduction  : 

1.  Tan  shoes  from  $5  to  $3.75. 

2.  Leather  belts  from  15  ^  to  5  ^. 

3.  Couches  from  115  to  110. 

4.  Screen  doors  from  $  1  to  29  ^. 

6.  Window  screens  from  25  ^  to  10  ^. 

6.  Rockers  from  $2.50  to  $1.70. 

7.  Handkerchiefs  from  15^  to  9^. 

8.  Sheeting  unbleached  from  6^  to  3^. 

9.  Bicycles  from  $  25  to  $  16.50. 

10.  Boys'  bicycles  from  $  20  to  $  14.50. 

11.  Bicycle  shoes  from  $  2.50  to  $  1.25. 

12.  Shoes  from  $  3  to  $  1.85. 

13.  Veils  from  $5  to  $2.25. 

14.  Dress  skirts  from  $  8  to  $  5.75. 

15.  Bookcases  from  $  12.50  to  $  7. 

16.  Towels  from  15  ^  to  121  ^. 

17.  Towels  from  50  ^  to  35  ^. 

18.  Towels  from  $  75  ^  to  50  ^ 


112  MENTAL  ARITHMETIC 

19.  Curtains  from  $  2^  to  $  IJ 

20.  Skirts  from  40  ^  to  25 1 

21.  Blankets  from  $  5  to  $  3f . 

22.  Comforters  from  $  4  to  1 2.98. 

23.  Comforters  from  $  SJ  to  $  21 

24.  Laces  from  |  \\  to  25  ^. 

25.  Jackets  from  $  20  to  $7|. 

Lesson  38 

What  fractions  in  their  lowest  terms  are  equiva- 
lent to  the  following  per  cents  ? 

1.  \%  2%,  4%,  5%,  20%,  25%,  50%,  60%, 
100  %  ? 

2.  75  %,  45  %,  36  %,  90  %,  18  %,  32  %,  56  %  ? 

3.  121  %,  621  %,  871  %,  371  %,  331  %,  66|  %  ? 

4.  16f  %,  831%,  81  %,  111  %,  x^%  6f  %,  6J%  ? 

5.  93-\%,44i%,  5f%,  88f%,  27|%? 

6.  100  %,  120  %,  125  %,  225  %,  350  %,  475  %  ? 

7.  A  lot  which  cost  $600  was  sold  at  a  gain  of 
40  % .  The  gain  is  equal  to  what  part  of  the  cost  ? 
What  was  the  gain  ?     The  selling  price  ? 

8.  Cloth  which  cost  42^  a  yd.  was  sold  at  a  loss 
of  16|%.  The  loss  was  what  fraction  of  the  cost? 
What  was  the  loss  on  each  yard?  The  selling 
price  ? 


LESSON  38 


113 


9.  Silk  which  cost  $3.60  a  yd.  was  sold  at  an  ad- 
vance of  33|^%.  Find  the  gain  and  the  selling  price 
per  yd. 

10.  A  farmer  who  had  360  sheep  sold  22|%  of 
them.  How  many  did  he  sell  ?  How  many  did  he 
keep? 

11.  The  following  articles  were  advertised  in  the 
papers  subject  to  the  discounts  given.  Find  the 
actual  selling  price : 


Marked  price 

Discount 

1.    Tan  shoes    . 

.     .  $  6.00 

25   % 

2.    Lace  curtains 

.     .  f  2.00 

37-1% 

3.    Capes     .     . 

.     .  1  7.50 

33  J  % 

4.    Shirts     .     . 

.     .  1  1.25 

60   % 

5.    Towels  .     . 

.     .  1  0.50 

30   % 

6.    Towelling  . 

.     .  *  0.121 

40   % 

7.    Parlor  suits 

.  f42.50 

60   % 

8.    Linen  skirts 

.     .  1  4.00 

621% 

12.    Find  the  selling 

price  of  the  following  articles 

subject  to  the  discount 

s  given : 

M 

\RKED  PRICK 

Discount 

1.    Silks      .     . 

$  1.75 

14f% 

2.   Tables  .     . 

f  4.50 

16f% 

3.    Chairs  .     . 

f  0.75 

20   % 

4.    Couches 

112.00 

121% 

5.    Umbrellas  . 

1  1.98' 

14% 

6.    Suits      .     . 

$15.00 

10   % 

114  MENTAL  ARITHMETIC 

13.  A  farmer  sold  his  crop  of  wheat  for  1896  for 
$180.  The  next  year  his  wheat  crop  was  |  greater, 
and  the  price  of  wheat  33J%  more  than  in  1896. 
Find  what  he  got  for  the  crop  of  1897. 

14.  What  is  the  selling  price  of  an  article  costing 


116  and  sold 

.  at  a  loss  of  \^%1 

At  a  gain  of 

12|% 

? 

15. 

What 

part   of    1  %    is 

j%? 

\1o1 

i%? 

f%? 

f%? 

A%? 

16. 

What  : 

is  1%  of  $1800/ 

'  i%? 

i%? 

l%? 

!%? 

f%? 

17. 

If   \1o 

is  charged  for 

sending 

'  money 

from 

New  York  to  Boston,  what  is  charged  for  sending 
$800?     $600?     $300?     $500? 

18.  If  1^  %  commission  is  charged  for  selling  stock, 
what  commission  is  charged  for  selling  $1600  stock  ? 
$2400?    $848?     $3264? 

19.  What  is  the  cost  of  insuring  600  bbl.  of  flour 
worth  $5  per  bbl.,  the  cost  of  insurance  being  \%  oi 
the  value  of  the  flour  ? 

20.  A  merchant  paid  60^  a  yd.  for  cloth  and  sold 
it  at  an  advance  of  33^%.  After  selling  all  but 
2  yd.,  he  put  the  remainder  on  the  remnant  counter 
and  marked  it  to  sell  at  50  %  below  the  first  selling 
price.     Find  what  it  was  marked  at. 

21.  A  grocer  bought  potatoes  at  45^  a  bushel  and 
sold  them  at  a  gain  of  33^  % .     If  he  received 
for  the  potatoes,  how  many  bushels  did  he  sell  ? 


LESSON  39  115 

22.  A  fruit  dealer  paid  $10  for  peaches  at  25^  a 
basket,  part  of  which  being  spoiled  he  sold  the  re- 
mainder at  an  advance  of  60  %,  receiving  |12.  How 
many  baskets  were  damaged? 

23.  A  lawyer  collected  |320  and  charged  5%  for 
his  services.  How  much  did  he  retain  and  how 
much  did  he  pay  over  to  his  employer  ?  What 
per  cent  is  the  amount  paid  over  of  sum  collected  ? 

24.  A  bookseller  paid  80)^  each  for  some  books. 
The  expense  of  selling  them  was  5  %  of  the  cost,  and 
the  gain  16|  %  of  the  whole  outlay.  Find  their  sell- 
ing price. 

25.  A  grocer  mixes  two  kinds  of  tea,  which  cost 
him  44^  and  66^  respectively,  in  equal  quantities. 
What  must  be  the  selling  price  of  the  mixture  to 
gain  20  %  ? 

Lesson  39 

1.  A  house  worth  $2500  rents  for  $300.  What 
is  the  ratio  of  the  rent  to  the  value  of  the  house  ? 
For  what  per  cent  of  its  value  does  it  rent  ? 

2.  A  town  of  2400  population  increases  to  2700 
in  one  year.  What  is  the  ratio  of  the  increase  to 
the  original  population  ?  Find  the  rate  per  cent  of 
increase. 

3.  A  drover  bought  sheep  at  $  5  each ;  their  food 
cost  $1.25  apiece.  He  then  sold  them  at  $7.50  each. 
Find  his  gain  per  cent. 


116  MENTAL  ARITHMETIC 

4.  A  man  invested  §600  in  real  estate  and  sold 
out  at  a  gain  of  331%.  He  re-invested  the  entire 
amount  and  again  sold  out  at  an  advance  of  25%. 
Find  the  last  selling  price  and  the  total  gain. 

5.  A  trader  sold  a  horse  at  an  advance  of  22|%, 
gaining  |16.  This  22|  %  is  22|  %  of  what  ?  What 
fraction  of  the  cost  ?  What  was  the  cost  of  the 
horse  ? 

6.  A  grocer  made  $12  by  selling  flour  at  a  gain  of 
16|  % .     What  did  it  cost  him  ? 

7.  A  merchant  lost  $18  by  selling  cloth  at  a  loss 
of  37|^  %.     Find  what  he  paid  for  it. 

8.  I  was  offered  $800  for  a  lot,  but  sold  it  for 
$  824,  gaining  4  %  more  than  if  I  had  accepted  the 
first  offer.     Find  what  the  lot  cost  me. 

9.  A  merchant  sold  tea  at  64  ^  a  lb. ,  gaining  33|^ % . 
The  gain  is  what  fraction  of  the  cost  ?  The  selling 
price  is  what  fraction  of  the  cost  ?  How  much  a 
pound  did  the  tea  cost  the  merchant  ? 

10.  A  man  sold  a  cow  for  $36,  gaining  33^%. 
Find  the  cost. 

11.  What  would  have  been  the  cost  price  of  the 
cow  given  in  example  10  if  there  had  been  a  gain  of 
20%?    Of  50%?    121%?    80%?    93-1-%?    111%? 

12.  A  grocer  sold  coffee  at  28^  a  lb.,  thereby 
losing  121%.  The  loss  is  what  fraction  of  the 
cost  ?  The  selling  price  is  what  fraction  of  the 
cost?     What  was  the  cost? 


LESSOi^  3d  117 

13.  A  merchant  sold  cloth  at  48i^  a  yard,  thereby 
losing  331  ^ .     What  did  it  cost  him  ? 

14.  What  would  have  been  the  cost  price  of  the 
cloth  in  example  13  if  there  had  been  a  loss  of  25  %  ? 
Of   20  %?       142%?       111%'/       331%?       7_9_%? 

40%?      60%?- 

15.  Eggs  sold  at  12  j^  a  doz.  net  a  profit  of  20  %  ; 
find  the  cost  price  per  doz. 

16.  By  selling  clothes  at  112  a  suit  a  merchant 
gains  20  %.  What  would  have  been  the  actual  gain, 
and  what  the  gain  per  cent,  if  they  had  been  sold  for 
$12.50  a  suit? 

17.  A  merchant  sells  overcoats  at  f  21,  thereby 
losing  12J%.  What  was  the  cost,  and  what  per 
cent  would  he  gain  by  selling  them  at  $28  apiece? 

18.  A  wholesale  merchant  sold  goods  to  a  cus- 
tomer at  a  profit  of  25%,  but  the  buyer  becomes 
bankrupt  and  pays  only  60  j^  on  the  dollar.  What 
per  cent  does  the  wholesale  merchant  lose  on  the 
sale  ? 

19.  A  merchant  bought  goods  for  $900.  He  sold 
I  of  them  at  a  gain  of  15  %  and  the  remainder  at  a 
loss  of  24%.  Find  his  gain  or  loss  on  the  whole 
transaction. 

20.  A  grocer  sold  a  chest  of  tea  at  69 j^  a  lb., 
thereby  gaining  15  %.  If  his  entire  gain  was  $6.75, 
how  many  pounds  of  tea  did  the  chest  hold? 


118  MENTAL  ARITHMETIC 

21.  I  sold  a  span  of  horses  at  $105  each,  gaining 
16|%  on  one  and  losing  16|%  on  the  other.  Find 
the  cost  price  of  each  horse  and  the  gain  or  loss  on 
both  transactions. 

22.  A  speculator  sold  two  lots  at  $720  each,  los- 
ing on  one  20%  and  gaining  20%.  on  the  other. 
Find  the  gain  or  loss  on  the  whole  transaction. 

23.  How  many  yards  of  cloth  at  $3  a  yd.  must  a 
merchant  buy,  so  that  by  selling  it  at  12|  %  profit 
he  gains  $24? 

24.  Give  answers  to  the  following  : 
(ti)  18|%  of  48  =  ? 

(5)  $480  is  18f  %  of  what  number? 
(c?)   .8%  of  20  =  ? 
(J)  20  is  .8  of  what  number? 
(^)   ^  is  what  per  cent  of  f  ? 

25.  A  grocer  buys  flour  at  the  rate  of  $5.40  a  bbl. 
At  what  price  per  49-lb.  sack  must  he  sell  it  so'as  to 
gain  111  %  ? 

Lesson  40 

1.  A  wholesale  merchant  sold  a  bill  of  goods,  list 
price  $240,  discounts  16|%  and  10  %.  Find  the  net 
amount  of  the  bill. 

2.  What  is  the  net  amount  of  a  bill  of  goods,  the 
list  price  of  which  is  $  300,  trade  discounts  20  %  and 
121  %  ?  What  would  have  been  the  net  amount  with 
trade  discounts  12|  %  and  20  %  ? 


LESSON  40  119 

3.  What  single  discount  would  give  the  same  net 
amount  for  a  bill  of  $300  as  discounts  of  20%  and 

4.  Find  the  net  cash  amount  of  a  bill  for  1400, 
subject  to  discounts  of  25%,  16|%,  and  4%  off  for 
cash. 

5.  What  is  the  difference  between  the  net  amount 
of  a  bill  of  goods  for  $800,  trade  discounts  25%  and 
15  %,  or  a  single  discount  of  40  %  ? 

6.  A  merchant  buys  at  40  %  below  the  list  price, 
and  sells  at  20  %  below  the  list  price.  Find  his  gain 
per  cent. 

7.  An  auctioneer  sold  goods  on  a  commission  of 
2%.  What  did  he  receive  on  a  sale  of  $1400?  What 
per  cent  of  the  sales  does  his  employer  receive? 

8.  A  commission  merchant  received  2J%  for 
selling  goods.  If  his  commission  amounted  to  $75, 
what  was  the  selling  price? 

9.  A  broker  sold  $  6000  stock  on  a  commission  of 
^%.     Find  his  commission. 

10.  A  commission  merchanf  sold  600  bbl.  of  flour 
at  $4.50  on  a  commission  of  3  %.  Find  the  commis- 
sion and  the  amount  remitted  to  his  employer. 

11.  My  agent  received  $12  for  selling  apples  at 
$2.50  a  barrel  on  a  commission  of  4%.  How  many 
barrels  did  he  sell  ? 


120  MENTAL  ARITHMETIC 

12.  I  paid  my  insurance  premium  of  $50  with  J  of 
my  commission  at  6  %  for  selling  land.  For  how 
much  did  the  land  sell  ? 

13.  A  young  man  was  paid  6  %  commission  by 
a  merchant  for  collecting  overdue  accounts.  If  he 
earned  $  31.80,  how  much  did  he  collect  ? 

14.  A  man  insured  his  house  for  $  1875  at  2  % 
premium.     How  much  did  he  pay  ? 

15.  I  insured  my  furniture  for  three  years  for 
$600  at  1^%  premium.  What  was  the  amount  of 
the  premium  ? 

16.  A  mill  owner  insured  his  lumber  for  $  800, 
paying  $25  premium.  What  rate  of  insurance  did 
he  pay  ? 

17.  An  insurance  company  received  $22.50  for 
insuring  a  house  for  75%  of  its  value  at  |%  pre- 
mium.    Find  its  value. 

18.  A  farmer  paid  $  6.25  for  insuring  his  barn  and 
contents  at  |  % .     For  what  amount  did  he  insure  ? 

19.  A  merchant  insured  his  stock  for  $400  for  one 
year  at  |  % .  Six  months  after  the  policy  was  can- 
celled at  the  request  of  the  insured.  Find  the 
amount  of  premium  returned,  the  short  rate  for  six 
months  being  |%. 

20.  I  paid  $  27  for  insuring  my  house  at  |  of  its 
value  at  1^ %.     What  is  the  value  of  the  house? 


LESSON  41  121 

21.  My  furniture  was  insured  in  one  company  for 
$600  and  in  another  for  1 400.  It  was  damaged  to 
the  extent  of  $  30.  For  what  amount  was  each  com- 
pany liable  ? 

22.  My  property  is  assessed  for  $  7200.  What  tax 
must  J  pay,  the  rate  of  taxation  being  1^  %  ? 

23.  What  amount  must  a  town  be  taxed  to  raise 
$  11,400,  the  collector's  commission  being  5  %  ? 

24.  The  rate  of  taxation  being  reduced  from  19 
mills  to  17  mills  on  the  dollar,  my  taxes  are  lowered 
$  8.     For  how  much  am  I  assessed  ? 

Lesson  41 

Find  the  interest  of  : 

1.  $600  for  1  yr.  at  5%;  4%;  6%;  8%. 

2.  $400for  lyr.  at  7%;  3%;  9%;  10%. 

3.  $800for  lyr.  at4%;  41%;  61%;  31%. 

4.  |450for  lyr.  at4%;  5%;  8%;  9%. 

5.  $250  for  lyr.  at  3%;  31%;  51%;  71%. 

6.  $375for  lyr.  at4%;  3%;  5%;  8%. 

7.  6  %  per  annum  is  what  per  cent  for  6  mo.? 
4  mo.?     8  mo.?     3  mo.?     9  mo.?     10  mo.? 

8.  8  %  per  annum  is  what  per  cent  for  3  mo.? 
6  mo.?     4  mo.?     4J  mo.?     9  mo.? 

Find  the  interest  of : 

9.  $  500  for  6  mo.  at  8  %  ;  3  mo. ;  9  mo. ;  4  mo. 
10.    $  700  for  6  mo.  at  6  %  ;  10  mo. ;  4  mo. ;  9  mo. 


122  MENTAL  ARITHMETIC 

11.  1 350  for  6  mo.  at  4  %  ;  9  mo. ;  3  mo. ;  4  mo. 

12.  What  part  of  a  year  is  2  mo.  12  da.?  1  mo. 
15  da.?  9  mo.  18  da.?  6  mo.  12  da.? 

Find  the  interest  of  : 

13.  1 640  for  2  mo.  12  da.  at  5  %  ;  7  mo.  6  da. 

14.  $  325  for  6  mo.  12  da.  at  TJ  %  ;  9  mo.  18  da. 

15.  1400  for  3  mo.  15  da.  at  6  %  ;  10  mo.  15  da. 

16.  In  each  of  the  above  questions  in  which  you 
find  the  interest,  how  would  you  find  the  amount? 
Find  the  amount  for  the  first  part  of  each  of  these 
questions. 

17.  A  merchant  whose  capital  was  f  15,000  was 
making  a  profit  of  12^%  per  annum  on  his  capital, 
but  on  account  of  ill  health  he  quit  the  business,  and 
loaned  his  money  at  6  %.  How  much  is  his  yearly 
income  diminished  ? 

18.  A  man  leaves  unpaid  a  sum  of  money  on 
which  he  pays  8  %  interest  until  the  interest  equals 
-^Q  of  the  principal.     Find  the  time. 

19.  A  person  borrows  a  sum  of  money  for  9  mo. 
at  6  %,  and  at  the  end  of  the  time  repays,  as  princi- 
pal and  interest,  $418.     How  much  did  he  borrow  ? 

20.  Feb.  15,  1898,  a  young  man  deposited  in  the 
savings  bank  |450.  July  15,  1898,  he  withdrew  the 
principal  and  interest  at  4  %  per  annum.  What 
amount  did  he  withdraw  ? 


LESSON  42  123 

21.  I  drew  a  note  for  $  550,  due  in  6  mo.,  and  pre- 
sented it  at  a  bank  for  discount  at  8  % .  What  was 
the  discount  ?     What  were  the  proceeds  of  the  note  ? 

22.  What  is  the  bank  discount  on  a  note  for  $800 
due  in  1^  mo.  at  6  %?  How  much  cash  would  the 
owner  of  this  note  receive  for  it  ?  Does  the  bank 
make  6  %,  or  more  or  less  than  6  %,  on  this  note  ? 

23.  A  note  due  in  3  mo.  was  presented  at  a  bank, 
and  when  discounted  at  8  %  yielded  $  490.  Eequired 
the  face  of  the  note. 

24.  A  wholesale  merchant  took  a  note  due  in 
4  mo.  in  payment  of  certain  goods.  He  had  it  dis- 
counted immediately  at  9%,  and  received  $582. 
For  what  sum  were  the  goods  sold  ? 

25.  A  merchant  sold  goods  for  $  540,  subject  to 
trade  discounts  of  33^  %  and  16|  %,  and  took  in  pay- 
ment a  note  due  in  4  mo.,  which  he  had  immediately 
discounted  at  8  %.     Find  the  cash  value  of  the  sale. 

Lesson  42 

1.  If  I  buy  rugs  at  $4.90  and  sell  them  for  $  5.60 
each,  what  is  my  gain  per  cent  ? 

2.  A  retail  merchant  bought  carpet  at  $1.55  list 
price,  trade  discount  20%,  and  sold  it  at  an  advance 
of  25  %  on  cost.     Find  the  selling  price. 

3.  If  331  %  is  gained  by  selling  cloth  at  $  1.00  a 
yd.,  find  the  gain  or  loss  per  cent  if  sold  at  80)^  per  yd. 


124  MENTAL  ARITHMETIC 

4.  A  tradesman's  prices  are  40  %  above  cost.  If 
he  allows  a  customer  10%  on  his  bill,  what  profit 
does  he  make  ? 

5.  A  person  at  the  age  of  25  insured  his  life  in 
each  of  two  companies  for  $4000,  the  premiums  being 
at  the  rate  of  2^  %  and  2i  %  respectively.  Find  his 
annual  payment. 

6.  A  trader  insured  300  bbl.  of  flour  for  |  of  its 
value  at  2%,  paying  $25  premium.  What  did  the 
flour  cost  him  a  barrel  ? 

7.  A  man  whose  property  is  assessed  for  f  12,000 
is  taxed  15  mills  on  the  dollar.     Find  his  tax. 

8.  A  town  levied  a  tax  to  build  a  town  hall  at  a 
cost  of  $4900.  If  the  collectors  fee  is  2  %,  find  the 
tax. 

9.  I  bought  goods  for  $400,  list  price.  Is  it  to 
my  advantage  to  accept  trade  discounts  of  25  and  5 
or  of  20  and  10? 

10.  Goods  are  sold  at  a  loss  of  20%.  By  what 
percentage  of  itself  must  the  selling  price  be  advanced 
to  yield  a  profit  of  20  %  ? 

11.  A  man  began  business  with  a  certain  capital ; 
he  gained  20  %  the  first  year,  which  he  added  to  his 
capital,  and  37|  %  the  second  year,  which  he  also 
added  to  his  capital.  If  his  capital  at  close  of  the 
second  year  was  $6600,  what  was  his  original  capital? 


LESSON  42  125 

12.  A  man  began  business  with  a  certain  capital ; 
he  gained  10%  the  first  year,  which  he  added  to  his 
capital,  and  20%  the  second  year,  which  he  also 
added  to  his  capital ;  the  third  year  he  lost  9  Jy  % . 
He  then  had  11000  more  invested  in  the  business 
than  at  first.  Find  the  capital  with  which  he  com- 
menced business. 

13.  A  grocer  bought  60  lb.  of  tea  and  90  lb.  of 
coffee,  the  cost  of  the  latter  per  lb.  being  331  %  that 
of  the  former.  He  sold  the  tea  at  a  profit  of  25%, 
and  the  coffee  at  a  profit  of  16|%,  gaining  on  the 
whole  $14.40.  Find  the  buying  and  selling  prices 
of  each. 

14.  A  carpet  dealer  sold  carpet  at  98^  a  yd.,  gain- 
ing 16|%.  Afterward  he  sold  some  of  the  same 
carpet  to  the  amount  of  $28.80  and  gained  14|%. 
What  was  the  last  selling  price  and  how  many  yards 
did  he  sell  at  that  price  ? 

15.  A  merchant  sold  40  yd.  of  silk  at  a  gain  of 
Sl^  %,  and  45  yd.,  which  cost  him  the  same  per  yard, 
at  a  gain  of  6|%.  If  his  entire  gain  was  $36,  find 
the  cost  price  per  yard. 

16.  A  merchant  bought  goods  for  a  certain  sum, 
and  marked  |  of  them  at  a  profit  of  25%,  and  the 
remainder  at  a  profit  of  15%.  If  his  entire  gain  is 
$76.38,  find  what  he  paid  for  the  goods. 

17.  A  stationer  buys  a  gross  of  lead  pencils  for 
$2.50.  He  sells  44  at  the  rate  of  2  for  5^  and  the 
remainder  at  3^  each.     Find  his  gain  per  cent. 


126  MENTAL  ARITHMETIC 

18.  A  grocer  purchases  syrup  in  50  gal.  barrels  at 
$  12.50  a  bbl.  At  what  price  per  gallon  must  he  sell 
it  to  make  16  %  ? 

19.  I  bought  a  house  for  $  3000  and  spent  331  ^  of 
the  cost  in  repairs.  What  must  I  rent  it  for  a  month 
to  make  a  clear  income  of  6  %  of  the  total  cost,  taxes 
and  repairs  amounting  to  f  60  a  year  ? 

20.  A  man  having  lost  6^  %  of  his  capital  is  worth 
twice  as  much  as  another  who  has  just  gained  50  %  on 
his  capital.     What  was  the  first  man's  capital? 

21.  If  a  furniture  dealer  buys  chairs  at  a  discount 
of  30  %  from  list  price,  and  sells  them  at  9  %  below 
list  price,  find  his  gain  per  cent. 

22.  A  certain  sum  of  money  amounts  to  $  1236  in 
6  mo.,  and  to  $1254  in  9  mo.  What  is  the  rate  per 
cent? 

23.  Bought  2000  bu.  of  wheat  at  82^  a  bu.,  pay- 
able in  4  mo.;  I  sold  it  immediately  for  80^  a  bu. 
cash,  and  put  the  money  at  interest  at  6  % .  At  the 
end  of  4  mo.  I  paid  for  the  wheat.  Did  I  gain  or 
lose  by  the  transaction,  and  how  much  ? 

24.  A  wholesale  merchant  bought  400  yd.  of  silk 
at  $2.50  a  yd.,  and  sold  it  at  once,  receiving  in  pay- 
ment a  note  for  $1100  due  in  60  da.,  which  he  at 
once  discounted  at  a  bank  at  6  % .     Find  his  gain. 


LESSON  43  127 

MISCELLANEOUS  EXAMPLES 

Lesson  43 

1.  How  many  strips  of  paper  2  ft.  wide  are  needed 
to  cover  the  walls  of  a  room  whose  length  is  18  ft. 
and  whose  breadth  is  f  its  length  ? 

2.  A  gardener  sold  7  bu.  of  potatoes  at  35^  a 
bushel,  and  with  the  proceeds  he  bought  3  yd.  of 
ribbon  at  Wf^  a  yard  and  sugar  at  5)^  a  lb.  How 
many  pounds  of  sugar  did  he  buy  ? 

3.  A  township  is  cut  into  farms  of  80  A.  Find 
the  number  of  farms. 

4.  Two  men  separate,  one  driving  north  at  the  rate 
of  7  mi.  an  hr.  and  then  returning  at  the  same  rate. 
The  second  walks  south  at  the  rate  of  3  mi.  an  hr. 
How  far  apart  will  they  be  in  3  hr.? 

5.  The  distance  of  a  suburb  of  Chicago  from  the 
city  is  indicated  in  any  one  of  the  following  ways : 
11  mi.,  or  24  min.,  or  $5  for  a  monthly  ticket. 
Which  unit  gives  the  most  practical  measurement  of 
the  distance  of  the  suburb  from  the  city  ? 

6.  Harriet  puts  68  p  into  the  Penny  Savings  at  one 
time,  and  has  stamps  of  the  largest  denomination 
possible  placed  on  her  card  as  a  receipt.  If  1  ^,  5  ^, 
10^,  and  25  p  stamps  are  used  for  the  purpose,  how 
many  of  each  kind  are  used  ? 

7.  Dorothy,  who  has  7  10^  stamps,  3  5^  stamps, 
and  9  1^  stamps  on  her  penny  savings  card,  has  it 
cashed.     How  much  does  she  realize  ? 


128  MENTAL  ARITHMETIC 

8.  The  annual  membership  fee  for  adults  in  the 
Riverside  Art  League  is  50^,  and  for  children  10^. 
What  will  a  family  of  two  adults  and  three  children 
pay  in  dues  in  3  yr.  if  all  are  members  ? 

9.  What  is  the  total  cost  of  a  revolver  weighing 
1^  lb.,  for  which  $9.50  was  paid,  if  sent  through  the 
mail  in  a  registered  parcel,  the  buyer  paying  postage 
and  the  cost  of  registration  ? 

10.  I  bought  two  crates  of  currants  at  different 
times  and  made  28  glasses  of  jelly  out  of  each.  The 
jelly  from  the  first  crate,  for  which  I  paid  90  ^,  cost 
a  cent  a  glass  more  than  that  made  from  the  second 
crate.     Find  what  the  second  crate  cost  me. 

11.  La  Grange  is  13  mi.  west  of  Chicago,  and 
Niles  60  mi.  east.  How  far  apart  are  two  places, 
one  of  which  is  7  mi.  east  of  La  Grange,  and  the 
other  8  mi.  west  of  Niles  ? 

12.  The  sum  of  the  ages  of  three  brothers  is  57  yr. 
The  youngest  is  15  yr.  old  and  the  next  2  yr.  older. 
Find  the  difference  between  the  ages  of  the  oldest 
and  the  youngest. 

13.  A  lady  goes  to  the  city  by  train  at  the  cost  of 
$3.10  for  a  25-ride  ticket  and  a  5^  street-car  fare 
each  way,  using  4  25-ride  tickets  in  one  year.  If  she 
had  gone  by  electric  car  it  would  have  cost  her  10  ^ 
each  way.  How  much  would  she  have  saved  in  one 
year  if  she  had  gone  by  the  electric  car  ? 


LESSON  43  129 

14.  If  in  the  previous  question  it  takes  36  min.  to 
go  by  train  and  street  car  and  1  hr.  by  electric  car,  how 
many  hours  are  saved  in  one  year  by  going  by  train  ? 

15.  The  sides  of  three  boxes,  each  4  ft.  long,  3  ft. 
wide,  and  1  ft.  6  in.  high,  are  covered  with  wall 
paper  cut  from  the  same  double  roll.  How  much  is 
left  over  if  there  is  no  waste  ? 

16.  Alice  has  8  oranges,  Caryl  6,  and  Marguerite 
1.  The  oranges  are  divided  equally  amongst  them, 
and  Marguerite  pays  Alice  and  Caryl  16^.  How 
much  should  each  receive  ? 

17.  When  is  a  number  divisible  by  2  ?  By  4  ?  8  ? 
5?  10? 

18.  A  lady  buys  stockings  at  the  rate  of  3  pairs 
for  $1  instead  of  35^  a  pair.  How  many  pairs  of 
stockings  will  she  buy  before  she  saves  the  price  of  a 
single  pair  ? 

19.  The  average  age  of  10  boys  is  8  yr.  Five  boys 
join  the  group  whose  average  age  is  3  yr.  below  the 
average  of  the  others.  What  is  the  average  of  the 
entire  group  ? 

20.  A  barn  is  32  ft.  wide  and  its  two  gables  are  12 
ft.  above  the  eaves.  How  long  are  the  rafters  if  they 
project  1  ft.  6  in.  over  the  sides  ? 

21.  Two  men  start  from  the  same  point  and  walk, 
one  south  at  the  rate  of  3  mi.  an  hr.,  and  the  other 
west  at  the  rate  of  4  mi.  an  hr.  How  far  will  they 
be  apart  in  3  hr.  ? 


130  MENTAL  ARITHMETIC 

22.  What  is  the  volume  of  a  cube  whose  surface 
contains  96  sq.  in.  ?  What  is  the  surface  of  a  cube 
8  times  as  large  ? 

23.  A  man  walks  across  the  diagonal  of  a  lot  8  rd. 
long  and  6  rd.  wide  4  times  a  day,  instead  of  walking 
around  the  corner  on  the  sidewalk.  How  much  time 
does  he  save  in  a  year  of  300  da.,  if  he  walks  at  the 
rate  of  3  mi.  an  hour  ? 

24.  A  man  spent  |  of  his  money  and  then  earned 
a  sum  equal  to  J  of  what  he  had  left.  He  then  had 
$  60  less  than  he  had  at  first.  How  much  did  he 
spend  ? 

25.  A  lady  spent  ^  of  her  money  for  a  pound  of 
tea,  and  with  the  remainder  she  bought  a  dozen 
lemons  at  3)^  each.     What  did  the  tea  cost  per  lb.  ? 

26.  A  train  ran  2  mi.  80  rd.  at  the  rate  of  36  mi. 
an  hour.  How  far  would  it  have  gone  in  1^  times  as 
long  at  the  rate  of  48  mi.  an  hour  ? 

27.  Sugar  worth  $  1.55  is  weighed  in  a  false  balance 
which  gives  only  15|^  oz.  to  the  pound.  What  is  the 
selling  price  of  the  sugar  ? 

28.  A  man  agreed  to  dig  a  cellar  for  f  12,  but  after 
doing  I  of  the  work  he  found  that,  owing  to  the 
nature  of  the  soil,  he  could  only  work  on  the  remainder 
|-  as  fast  as  he  expected.  What  should  he  have 
charged  for  the  entire  work  ? 

29.  Divide  $  30  between  two  men,  giving  the  first 
1 6  more  than  |  of  what  the  second  gets. 


LESSON  43  131 

30.  What  is  the  weight  of  1  cu.  ft.  of  water? 
What  is  meant  by  saj'iiig  that  the  specific  gravity  of 
mica  is  3  ?  What  is  the  weight  of  1  cu.  ft.  of  mica 
in  air  ?     In  water  ? 

31.  The  specific  gravity  of  lead  ore  is  7.4.  How 
many  pounds  will  1  cu.  ft.  of  lead  ore  weigh  in  water? 

32.  Magnetite  and  hematite  are  two  iron  ores  of 
specific  gravity  6.5  and  5.3  respectively.  How  much 
more  would  1  cu.  ft.  of  magnetite  weigh  than  1  cu. 
ft.  of  hematite  ?  How  much  more  would  it  weigh 
in  water  ? 

33.  A  gallon  of  water  weighs  about  S^  lb.  Into 
a  pail  full  of  water  a  piece  of  mica,  specific  gravity  3, 
is  carefully  put  and  1  gal.  of  water  overflows.  Find 
the  weight  of  the  piece  of  mica. 

34.  At  the  rate  of  how  many  feet  per  second  must 
air  be  forced  into  a  schoolroom  containing  40  persons, 
through  an  opening  whose  area  is  2  sq.  ft.,  in  order 
that  each  one  may  have  30  cu.  ft.  of  fresh  air  per 
minute  ? 

35.  A  vessel  can  sail  up  stream  at  the  rate  of  8.  mi. 
and  down  stream  at  the  rate  of  12  mi.  an  hour. 
How  far  can  it  go  down  stream  and  up  in  5  hr.  ? 

36.  The  parallel  sides  of  a  field  in  the  form  of  a 
trapezoid  are  respectively  60  rd.  and  100  rd.,  and  its 
depth  20  rd.  What  is  the  field  worth  at  $48.50  an 
acre  ? 


132  MENTAL  ARITHMETIC 

37.  A  rectangular  block  of  land  containing  24  A. 
is  80  rd.  on  one  side.  How  long  will  it  take  a  boy 
to  ride  around  the  block  on  his  bicycle  at  the  rate  of 
1  mi.  in  5  min.? 

38.  What  is  the  weight  of  the  wall  paper  required 
for  the  walls  of  a  square  room  15  ft.  long  and  9  ft. 
high,  allowing  12  sq.  yd.  for  doors  and  windows? 
(A  double  roll  of  wall  paper  is  16  yd.  long,  18  in. 
wide,  and  weighs  1|  lb.) 

39.  A  box  contains  150  lights  of  glass,  each  6  in. 
by  8  in.  How  many  lights  of  glass,  each  12  in.  by 
30  in.,  are  there  in  a  second  box  that  contains  the 
same  quantity  as  the  first  ? 

40.  Lubricating  oil  for  bicycles  is  sold  in  8-oz.  cans 
at  f  1.54  a  dozen  and  in  4-oz.  cans  at  $  1.10  a  dozen. 
By  what  per  cent  is  it  cheaper  to  buy  a  dozen  8-oz. 
cans  of  oil  than  to  buy  the  same  quantity  in  4-oz. 
cans  ? 

41.  The  cost  of  a  box  of  glass  6  in.  by  8  in.,  con- 
taining 150  lights,  is  $2.50,  and  the  cost  of  a  box 
12  in.  by  24  in.,  containing  25  lights,  is  13.25.  By 
what  per  cent  is  the  price  of  the  same  quantity  of 
glass  increased  in  ordering  the  larger  instead  of  the 
smaller  panes  ? 

42.  A  man  loans  f  of  his  money  at  6  %  and  the 
remainder  at  4|  %.  If  his  income  from  the  former 
is  $  250  more  than  from  the  latter,  what  is  his  capi- 
tal ? 


LESSON  43  133 

43.  Which  is  cheaper,  goods  bought  for  1 200  sub- 
ject to  discounts  of  25  %  and  20  %  or  to  discounts  of 
20%  and  25%? 

44.  I  bought  a  bill  of  goods  for  $240,  discounts 
20  %  and  16|%.  Find  my  gain  per  cent  on  selling 
them  for  $  235,  if  I  lose  $  15  in  bad  debts. 

45.  A  grocer  sold  |  of  his  potatoes  for  10  %  less 
t^  an  all  cost  him  and  the  remainder  at  28  %  below 
cost.     Find  his  gain  per  cent  on  the  entire  lot. 

46.  I  sold  a  lot  of  land,  receiving  for  it  1 800  cash 
and  a  note  for  |640  due  in  6  mo.,  which  was  imme- 
diately discounted  at  a  bank  at  6  % .  Find  the  cash 
value  of  the  selling  price. 


134 


MENTAL  ARITHMETIC 
Multiplication  Table 


Twice 

Three  times 

Four  times 

Five  times 

Six  times 

Seven  times 

1   is     2 

1   is     3 

1   is     4 

1   is     5 

1   is     6 

1   is     7 

2    "     4 

2    "     6 

2    "     8 

2   "   10 

2    "    12 

2   "   14 

3   "     6 

3   "     9 

3    "   12 

3   "   15 

3   "    18 

3    "   21 

4   "     8 

4   "   12 

4   "   16 

4    "   20 

4    "   24 

4    "   28 

5    *'   10 

5   "   15 

5   "   20 

6   "  25 

5    "   30 

5    "   35 

6    "    12 

6   "    18 

6    "   24 

6   "  30 

6    "   36 

6    "   42 

7    "   14 

7    "  21 

7    "  28 

7    "   35 

7    "   42 

7    "  49 

8   "   16 

8   "  24 

8   "   32 

8    "  40 

8    "   48 

8    "    56 

9   "'   18 

9   '*   27 

9   "   36 

9   "  45 

9    "   54 

9   "   63 

10   "  20 

10   "  30 

10   "  40 

10   "   50 

10    "   60 

10    "   70 

11    **   22 

11    "   33 

11    "   44 

11    "   55 

11    ''   66 

11    "   77 

12   "   24 

12    "   36 

12    "   48 

12    "   60 

12    "    72 

12    ''   84 

Eight  times 

Nine  times 

Ten  times 

Eleven  times 

Twelve  times 

1   is     8 

1   i 

s       9 

1   is     10 

1  is     11 

1   is     12 

2   "   16 

2    ' 

'     18 

2    "     20 

2    "     22 

2    "     24 

3   "   24 

3    ' 

'     27 

3   "     30 

3   "     33 

3   "     36 

4   "   32 

4   ' 

'     36 

4    "     40 

4   "     44 

4   "     48 

5   "  40 

5   * 

'     45 

5    "     50 

5    "     55 

5   "     60 

6    "  48 

6   ' 

'     54 

6    "     60 

6   "     66 

6   "     72 

7   "   56 

7   ' 

'     63 

7   "     70 

7    "     77 

7   "     84 

8   "   64 

8   ' 

'     72 

8   "     80 

8   "     88 

8   *'     96 

9    "   72 

9   ' 

'     81 

9   "     90 

9   "     99 

9   "   108 

10   "   80 

10   * 

'     90 

10   "    100 

10   ''   110 

10   ^'   120 

11    "   88 

11   * 

'     99 

11    ''   110 

11    "   121 

11    *'   132 

12   "  96 

12    ' 

'   108 

12   "   120 

12    "   132 

12   "   144 

TABLES  135 

UNITS  OF  VALUE 

United  States  Money 

10  mills  (m.)  =  1  cent  (ct.  or  f) 
10  cents  =  1  dime  (d.) 

10  dimes         =  1  dollar  (^) 
10  dollars       =  1  eagle  (E.) 

British  or  Sterling  Money 

4  farthings  (far.)  =  1  penny  (c?.) 
12  pence  =  1  shilling  (5.) 

20  shillings  =  1  pound  (£) 

5  shillings  =  1  crown 

21  shillings  =  1  guinea 


UNITS  OF  WEIGHT 

Avoirdupois  Weight 

16  ounces  (oz.)  =  1  pound  (lb.) 

100  pounds  =  1  hundredweight  (cwt.). 

20  hundredweight        =  1  ton  (T.) 
1  pound  Avoirdupois  =  7000  grains 
1  ounce  Avoirdupois  —  437J  grains 

Troy  Weight 

24  grains  (gr.)     =  1  pennyweight  (pwt.) 
20  pennyweights  —  1  ounce  (oz.) 
12  ounces  =  1  pound  (lb.) 

1  pound  Troy     =  5760  grains 

1  ounce  Troy      =    480  grains 


136  MENTAL  ARITHMETIC 

Apothecaries'  Weight 

20  grains  (gr.)  =  1  scruple  (9) 

3  scruples         =  1  dram  (3) 

8  drams  =  1  ounce  (  ^  ) 

12  ounces  =  1  pound  (lb.) 

1  pound  Apothecaries'  weight  =  5760  grains 

1  ounce  Apothecaries'  weight  =    480  grains 


UNITS  OF   LENGTH 
Long  Measure 

12  inches  (in.)  =  1  foot  (ft.) 

3  feet  =  1  yard  (yd.) 

5i  yards  or  16^  feet  =  1  rod  (rd.) 
320  rods  =  1  mile  (mi.) 

1  mi.  =  320  rd.  =  1760  yd.  =  5280  ft. 

Surveyors'  Linear  Measure 

100  links  (1.)  =  1  chain  (ch.) 
80  chains      =  1  mile  (mi.) 
1  ch.  =  4  rd.  =  22  yd.  =  66  ft.  =  792  in. 

UNITS  OF   SURFACE   OR  SQUARE   MEASURE 
Surface  or  Square  Measure 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 

9  square  feet  =  1  square  yard  (sq.  yd.) 

30 J  square  yards  =  1  square  rod  (sq.  rd.) 

160  square  rods  =  1  acre  (A.) 

640  acres  =  1  square  mile  (sq.  mi.) 

10  square  chains  =  1  acre ;  1  acre  =  4840  square  yards 
A  township  is  6  mi.  square,  and  contains  36  sq.  mi.     It  is 
divided  into  36  sections,  each  1  mi.  square. 


TABLES  137 

UNITS   OF   VOLUME 

Cubic  or  Volume  Measure 

1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.) 
27  cubic  feet  (cu.  ft.)      =1  cubic  yard  (cu.  yd.) 

A  cord  is  a  pile  8  ft.  long,  4  ft.  wide,  and  4  ft.  high.     It  con- 
tains 128  cu.  ft.     A  cubic  yard  of  earth  is  called  a  load. 


UNITS  OF  CAPACITY 

Liquid  Measure 

4  gills  (gi.)  =  1  pint  (pt.) 
2  pints  ==  1  quart  (qt.) 

4  quarts       =  1  gallon  (gal.) 

1  gallon  contains  231  cu.  in. 

Dry  Measure 

2  pints  (pt.)  =  1  quart  (qt.) 
8  quarts  =  1  peck  (pk.) 
4  pecks  =  1  bushel  (bu.) 

1  bushel  contains  2150.42  cubic  inches 

Apothecaries'  Fluid  Measure 

60  minims  (tt^)  =  1  fluid  dram  (f  3) 
8  fluid  drams  =  1  fluid  ounce  (f  5  ) 

16  fluid  ounces  =  1  pint  (O) 
8  pints  =  1  gallon  (cong.) 

1  minim  is  about  eqiial  to  1  drop 


138  MENTAL  ARITHMETIC 

UNITS   OF   TIME 

Measure  of  Time 

60  seconds  (sec.)  —  1  minute  (min.) 

60  minutes  =  1  hour  (hr.) 

24  hours  =  1  day  (da.) 

7  days  =  1  week  (wk.) 

365  days  =  1  common  year  (yr.) 

366  days  =  1  leap  year  (1.  yr.) 
100  years  =  1  century  (C.) 

February  has  28  or  29  days;   April,  June,  September,  and 
November,  30  days  each ;  the  other  months,  31  days  each. 

Angular  Measure 

60  seconds  (")  =  1  minute  (') 
60  minutes        =  1  degree  (°) 
360  degrees        =  1  circumference  (c.) 


MISCELLANEOUS  UNITS 


Numbers 
12  units  =  1  dozen  (doz.) 
12  dozen  =  1  gross 
12  gross   =  1  great  gross 
20  units   =  1  score 


Paper 
24  sheets     =  1  quire 
20  quires     =  1  ream 
2  reams     =  1  bundle 
5  bundles  =  1  bale 


Wheat,  beans,  clover  seed,  and  potatoes  each  weigh  60  lb.  to 
the  bushel ;  corn,  rye,  and  coarse  salt,  56  lb. ;  barley,  48  lb. ;  and 
oats,  32  lb. 


Public  School  Arithmetic 

BASED    ON 

McLellan  and  Dewey's 

"PSYCHOLOGY   OF   NUMBER" 

BY 

J.  A.  McLELLAN,  A.M.,  LL.D., 

AND 

A.  F.  AMES,  A.B. 


i2mo.    Strong  Buckram.    Price,  60  cents,  net. 


This  book,  based  upon  sound  psychological  principles,  stands  for  a 
needed  reform  in  the  methods  of  teaching  Arithmetic. 

SPECIAL    FEATURES. 

The  treatment  of  the  subject  is  in  strict  line  with  the  idea  of  num- 
ber as  measurement. 

This  true  idea  of  number  running  through  the  whole  work  estab- 
lishes the  unity  of  the  whole. 

Fractions  are  divested  of  their  traditional  difficulty  by  being  placed 
in  their  true  relation  to  integers. 

Great  care  has  been  taken  in  selecting  and  grading  the  examples. 

This  treatment  of  the  subject  will  prove  a  good  preparation  for 
algebra. 

THE    MACMTLLAN    COMPANY, 

66   FIFTH   AVENUE,   NEW   YORK. 


COMMENTS. 

**  I  can  see  that  it  is  an  important  contribution  to  the  art  of  teaching  numbers." 
—  W.  T.  Harris,  U.S.  Commissioner,  Bureau  of  Education. 

"  From  a  careful  examination  it  seems  to  me  to  have  many  advantages  over  the 
books  on  the  subject  now  in  use.  Its  wise  omission  of  topics  of  no  practical  use,  the 
clearness  of  its  methods  and  problems,  and  its  neat  typography  appeal  to  every 
teacher  who  has  occasion  to  deplore  the  bulky  and  involved  arithmetics  in  so  many 
of  our  schools."  —  George  Gilbert,  Principal  Chester  Academy,  Chester,  Pa. 

"I  heartily  approve  of  the  method  of  this  book."  —  W.  B.  Smith,  McDonogh 
School,  McDonogh,  Md. 

"  The  processes  are  explained  logically  and  the  subjects  are  arranged  in  their 
proper  order  in  the  course.  The  examples  given  are  such  as  require  thought  and  at 
the  same  time  are  not  such  as  can  be  considered  unfairly  puzzling,  nor  are  they  too 
simple,  but  are  all  such  as  might  be  learned  from  some  of  the  preceding  parts  of  the 
book."  —  Frederick  Doolittle,  Acting  School  Visitor  and  Clerk  of  Committee 
on  Educatiofi,  of  Connecticut. 

"  This  volume  is  a  very  successful  attempt  to  give  expression  to  the  better  teach- 
ing of  psychology  as  to  the  growth  and  development  of  the  idea  of  number.  The 
plan  of  the  work  is  thoroughly  scientific,  the  methods  are  well  presented  ana  the  ex- 
amples well  chosen.  This  little  book  should  assist  greatly  in  the  reform  in  teaching 
arithmetic,  now  in  progress."  —  Prof.  Alfred  I.  De  Lurv,  University  of  Toronto. 

"  This  book  contains  many  admirable  features.  I  like  especially  the  early  intro- 
duction of  decimal  operations."  —  Cyrus  Boger,  A.M.,  Superintendent  Schools, 
Lebanon,  Pa. 

"  Naturally  I  am  pleased  with  the  extent  to  which  the  book  bases  the  treatment 
of  fundamental  operations  of  fractions  and  ratio  upon  the  dea  of  measure  and  of 
numbers  as  units  of  measurement.  I  am  particularly  struck  with  the  fact  that  the 
pupil's  attention  is  definitely  called  to  some  special  quantity  or  whole  which  furnishes 
the  object  of  attention,  and  within  which,  so  to  speak,  the  numerical  processes  take 
place;  also  with  the  clearness  and  conciseness  of  the  method  of  treatment;  the  logi- 
cal order  of  the  selection  of  topics;  and  the  exclusion  of  useless  and  irrelevant  mat- 
ter. The  simplification  of  treatment  due  to  sticking  close  to  fundamental  principles, 
must  recommend  the  book  to  teachers  and  pupils  who  have  been  bewildered  by  the 
great  number  of  topics  treated  in  the  ordinary  arithmetic  —  topics  which  do  not 
differ  at  all  in  their  logical  or  arithmetical  basis,  but  are  simply  different  practical 
expressions  of  the  same  principle.  I  wish  the  book  the  success  it  deserves."  —  Pro- 
fessor Dewey,  University  of  Chicago. 

"  *  The  Psychology  of  Number,'  by  Professors  McLellan  and  Dewey,  placed  on  a 
rational  basis  the  methods  to  be  pursued  in  the  elementary  treatment  of  number. 
This  has  now  been  followed  by  *  The  Public  School  Arithmetic,'  by  Professors 
McLellan  and  A.  F.  Ames,  in  which  these  rational  methods,  set  forth  in  the  former 
work,  are  systematically  and  successfully  presented.  The  special  feature  of  this 
book  consists  in  its  treatment  of  number  as  the  result  of  measurement.  The  authors 
have  brought  out  very  clearly  the  proper  methods  of  dealing  with  the  fundamental 
operations,  with  fractions,  and  with  the  commercial  rules.  The  definitions  are  con- 
sistent and  accurate  —  a  feature  not  common  in  elementary  texts  of  aritlynetic. 
There  is  an  excellent  collection  of  well-graded  examples.  Few  of  the  tricky  prob- 
lems which  have  done  so  much  to  discredit  arithmetic  are  to  be  found.  The  book 
consequently  deserves  speedily  to  win  a  place  among  recognized  text-books."  — 
Professor  McKay,  McMaster  University. 

"One  of  those  wise  books  that  make  school  study  more  pleasant  and  effective 
than  in  the  old  days  when  routine  study  was  the  rule.  .  .  .  Both  teachers  and  pupils 
will  welcome  this  very  valuable  book."  —  Saturday  Evening  Gazette^  Boston. 


THE    MACMILLAN    COMPANY, 

QQ  FIFTH  AVENUE,   NEW  YORK. 


'  ui    OvJu^D 


c  , 


54Jr)26 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


